Since I started using the Android platform on my tablet last year, a Samsung Galaxy Note 10.1 (2012), I've never looked back in terms of what user made applications have to offer compared to those I've experienced from using an Apple device. So much so that recently I moved away from my Apple iPhone 4 to a Samsung Galaxy Note 3. I've also found myself mostly using the same set of applications on both Android devices.
Why am I writing this?
A fair portion of my family and friends also have a phone and/or tablet on the Android platform, and I've often made recommendations for various applications for a specific purpose.
Rather than trying to remember them all only to later forget saying something I've decided to put up a list of my favourite and most useful Android applications with a brief description of what they do and why I find them useful. That last point is important, as this is not a simple list of "OH HEY CHECK OUT THESE COOL APPS LULZ", no no, I use a ton of applications but this list is only for those I find useful to have nearly every day.
Some of them are for more advanced or specific technical purposes, and some also require your device to be rooted in order to function to their full potential or even at all. Rooting isn't something I'd recommend for those of you that aren't too tech-savvy, so if you're not confident doing that ignore the applications marked as only for rooted devices.
All applications are free unless I've specifically noted otherwise.
An alternative app store
So you're aware of Google's app store (Google Play) and possibly a few others (Samsung/Amazon), but there is also an app store called F-Droid. This store offers applications that are ONLY free and open source for Android devices. Thanks to Android being as open as it is, there is nothing stopping you using additional app stores (providing you choose to allow applications from "unknown sources"), where as an Apple device would have to be jailbroken to do this.
To use F-Droid you will need to download and install the store's package right here, assuming you find any applications on this page you want from there. You can use that package link directly on your device -- none of that copying stuff to your device manually malarky!
The applications
Enough with the boring intro, let's get to the good stuff.
Annoyance removals
Adblock Plus
Better on rooted devices.
This service application runs entirely in the background to monitor your web-based network traffic and filter out advertisments. It can filter adverts both in web pages and in other applications, meaning that many "pro ad-free" versions of applications are no longer needed. This was originally on Google Play but it got removed as it was deemed to violate the stores terms of use in that it modified or impaired the functionality of other apps, though it's still available via the F-Droid store.
Though your device does not need to be rooted to use it, this service will only be able to filter adverts over Wi-Fi connections on unrooted devices. This is because modifications to part of the core system need to be made in order to apply filtering to mobile data connections (2G/3G/4G).
Not every advert will be filtered. The service relies on a filtering list, of which is updated periodically. However these filter lists are shared from the popular Adblock Plus extension for many web browsers, so unfiltered adverts usually get a limited time before they are added to the filter. Oh yeah, I'd also recommend you get Adblock Plus web browser extension for your desktop/laptop computer too -- available for Internet Explorer, Mozilla Firefox, Google Chrome, Opera, and Safari.
Communication
K-9 Mail
K-9 is a replacement for the stock email application. Not only does it offer many interface improvements, it also offers support for a much broader range of mailboxes. Notifications (including the colour of the LED notifier) can all be configured on a per-mailbox basis, and it can even export all its' settings to a file so you can import them on another Android device.
This was very important for me as one of my mailboxes doesn't use the same folder structure as Google's Gmail. The stock application couldn't adapt to this, which is where K-9 came in. It also offers support for digitally signed and encrypted emails using an external security library, such as APG (an Android PGP library).
Textra SMS
Textra is a replacement for the stock SMS/MMS "Messages" application. Though it isn't massively different it does have a slightly cleaner interface, and notifications can be customized more so.
To be honest if you have no problem with the stock messages application you won't need this, I just prefer it.
Tiny Call Confirm
Ever clicked on someone's name in a text message, email, or somewhere only to see your phone is now calling them? You will surely know the panic and rush to end the call ASAP whilst cursing your phone in your head for doing this.
This tiny app will catch these accidental touches and produce a popup to confirm you really wanted to call the person. You do this by swiping a green phone icon aside to make the call, or touch a red hang up icon to cancel it as shown on the right.
There is something similar on Apple devices called AskToCall, but requires a jailbroken device to do so. No such hacks needed on Android! :)
Device maintenance & management
CCleaner
The popular (and excellent) computer cleaning utility from Pitform has been ported over to Android. This tool can look around your device's storage for unnecessary clutter and junk that you would be advised to remove.It won't remove your meaningful content like photos, music, videos, and such. By clutter/junk it means removing the stocks of data temporarily used by other applications in their day to day operation, of which can build up to consume a noticeable amount of space over time. You are of course shown exactly what will be removed before it does it, and you can opt for which junk you want it to remove. This tool is particularly useful for devices with a smaller amount of internal storage that may slow down or find applications not working properly when they are approaching their storage capacity.
This application may still in its' beta stage. Whilst its' still very usable and I would recommend it to anyone in its' current state, you need to opt-in for the beta version in order to obtain the application. You likely need a Google+ account (of which is just a couple of clicks to obtain as you likely already have a Google account), but don't worry, they don't send you any spam.
ES File Explorer / File Manager
Better on rooted devices.
This one is for the more advanced users that want a good file exploration and management application. It provides a good and clean interface for these jobs, and includes a useful set of features like a plain text editor, ZIP/RAR archive support, cloud storage support, and support for networked file systems such as bluetooth, FTP, SFTP (SSH file transfer), SMB/CIFS (Microsoft shared folders), and WebDAV.
The application can also manage the root filesystem of your device providing your device is rooted. There is also a sister app ES Task Manager, which sounds self-explanatory, and yes it is a good task/process manager.
All file management on my Android devices is done through this application now, as it just blows the stock file manager application out of the water. Particularly as I can access my stuff from my home file server anywhere in the world with it with its' built-in SMB/CIFS and SFTP clients.
Greenify
Better on rooted devices.
The more applications you install on your device over time, the more likely there are going to be applications running in the background slowing your device down and consuming its' battery.
This tool helps you identify applications using the most background processing power, of which both slow down your device and consume its' battery. It will then let you put those applications into a state of hibernation whilst they are not being used so your battery will last longer between charges.
Definitely useful if you're a Skype user like me, as that is a massive background battery muncher! It also pointed out other applications suitable for hibernation such as BBC iPlayer, eBay, Natwest, and Nectar. I'm not sure why those applications needed to do so much whilst not actively in use, but now they don't have to.
Titanium Backup
Only for rooted devices.
This application can make back up of literally everything about/on your device should it get erased accidentally or need to go through factory restore/recovery. Applications, their data/settings, your photos/music, you name it this can back it up.
All backups can be made in an incremental way too, meaning that only changes since the last made backup need to be done. This will drastically reduce the time subsequent backups take, depending on how often you do them. Backups can also be made automatically on a schedule, of which two schedules come pre-configured when you install the application.
If you decide to (and I recommend you do) purchase a pro version key you get a massive expansion of features, most of which I use are customizable schedules, cloud storage (as Google Drive and Dropbox both offer a lot of space for free) and backup encryption (should my external SD card or cloud storage get stolen/hacked). You can also schedule a task to sync your backup to cloud storage.
If you're rather paranoid about your device suddenly having a fit and needing a recovery like I am this application is very much my favourite choice of a backup. Do remember that your device has to be rooted for this application to be of any use at all.
Games
Simon Tatham's Puzzles
A collection of 34 single-player puzzle games available on Linux, OSX, and Windows has been ported to Android. Worth having for that moment of bordem.
Included games are Black Box, Bridges, Cube, Dominosa, Fifteen, Filling, Flip, Galaxies, Guess, Inertia, Keen, Light Up, Loopy, Magnets, Map, Mines (Minesweeper), Net, Netslide, Pattern, Pegs (Solitaire), Range, Rectangles, Same Game, Signpost, Singles, Sixteen, Slant, Solo (Sudoku), Tents, Towers, Twiddle, Unequal, and Untangle.
If for some reason you get an odd graphical glitch (usually the whole game suddenly slants to one side upon your first touch) it means you need to have your screen in vertical mode. This is a known issue.
Multimedia
MPC - Remote LITE
I use a Windows based PC to watch DVD's, TV shows, and movies as it has a 37" TV attached to it. Specifically with the MPC-HC application bundled with the popular K-Lite Codec Pack.
The TV being away from my desk though means I'd have to move over to it in order to pause the video, change it, etc. Not with this application I don't. It can fully control the MPC-HC application on my computer over Wi-Fi. Brilliant for a lazy lardwad like me.
There is an in-app purchase for a pro ad-free version, but providing you have installed AdBlock Plus you won't see the adverts.
MX Player
A video player of which supports a wide range of formats/codecs, makes use of hardware acceleration your device has to offer, and makes use of multiple processor cores. The result is a versatile video player with very smooth playback.
Not so much used on my phone, but good to have on my tablet. Combined with a file manager like ES File Explorer this application can make use of videos on remote file systems. In my case watching a video on my home server via SMB/CIFS (Microsoft shared folder).
This also comes with a range of sister applications of which offer additional codec support designed to work with the hardware your device has, so do check out what else the developer has to offer so you can get an additional package to make playback as smooth as possible.
Poweramp Music Player
This is the ultimate replacement for the stock music application. Where do I begin explaining the features of this? There are so many, so I'll just go over why I like it over the stock player.
Better sound quality (based on equalizers and tone adjustments), better interface, better playlist support, lower battery consumption, better lock screen, reliable background service process, and a good set of customizable home screen widgets.
This application isn't completely free mind. It gives you a 15 day trial (all features usable), after which you need to pay £2.49 to unlock it. Money very well spent indeed though!
Security & anti-theft
Google Authenticator
With our Google account becoming more used than ever before (Gmail, contacts, calendar, applications/music/anything on Google Play, 3rd party websites using your Google account as ID, and even to attach your Android device to your Google account in case you lose it) it has become pretty important to make sure your Google account is safe from misuse should your password get stolen or cracked. Have you ever tested how good a password of yours is or how long it would take to crack (on an average desktop PC)? No? Sort that out.
Once you've enabled 2-step authentication for your Google account it will be similar to the chip and pin we have with credit/debit cards. In order to sign in you need something you know (your password) and something you have (your phone). This makes your account so much harder to be used by someone else as both parts must be on hand to get inside.
Still unsure if this is for you? Watch this short video introducing Google 2-step authentication.
This application has also been opened up so that other services may take advantage the 2-step authentication benefits. Most notably in my case Dropox and various virtual currency exchanges I use now require me to verify a sign in with a fresh code my phone.
Prey anti-theft
Yes there are many applications out there that offer anti-theft features for your devices, such as remotely tracking and locking them. Why do we need another one? Even the antivirus application I recommend just below has this feature.
Prey differs from others by being purely dedicated and focused to the task of anti-theft. Recommended and featured by reputable organisations such as BBC news, The Washington Post, Huffington Post, Lifehacker, and TechCrunch for its' robust service and features purely for the task in hand. I'm not saying you should only use Prey, but rather there needs to be a good reason to not use it.
Such features other than the obvious tracking and locking include remotely taking a photo on the front/back cameras, triggering alarms even if your device is in silent mode, gathering network information for more accurate tracking, detecting SIM card changes, and activating Prey via SMS messages should a data network not be available. Prey can (and should) be granted administrative permission to your device, of which will prevent Prey being deactivated or uninstalled without your approval.
On a free Prey account you can protect up to 3 devices, be it Android, iOS, Linux, OSX, or Windows based. A premium account offers more features if you should need them, but for most of us a free account will do just fine.
TrustGo anti-virus & mobile security
A report by the reputable cyber-security company ESET shows that the Android mobile platform is still the most attacked compared to its' competitors (page 15) likely due to its' openness, and ability to install packages not from app stores. It is therefore a smart move to use a good security utility to help keep your device clean and scams away.
TrustGo will scan applications as you install them for known and possible threats, can alert you to known phishing scams as you browse websites, provides backup for your base data (contacts, call logs, and SMS/MMS messages), remotely locate your device in case of theft, take a photo of someone entering your screen PIN/password wrong 3 times, and much more.
Just like Prey above, this can (and should) be granted administrative permission to your device, of which will prevent it being deactivated or uninstalled without your approval.
Such a great set of features all for free, it has gained a lot of praise from users (160K out of 223K ratings are 5 stars) and has won numerous awards.
That's all for now
I hope this list has been somewhat useful to you, and thanks for reading. :)