Here are the apps I prefer to have on most Android phones.
Hangouts
It should be already built into your device, so no need to download and install it, but it'll be a powerful messenger app. It can also be programmed to receive SMS's and is constantly updated.
Root Explorer
This is a file manager dedicated to rooted devices, but if your device isn't rooted, don't worry, it still works wonders on non-root devices and I basically recommend it for Android.
VLC
Sometimes, videos may not open properly on the default video player. VLC will sometimes do wonders with these videos so get em'. As a sidenote, Android does not need that the videos be converted to a proper resolution and format before being played. If it can play it, it will play it.
Keep or OneNote
This time you'll choose between Google Keep or Microsoft OneNote, and both of these are dedicated to storing your notes in the cloud. Its quite nice if you want to copy a long note to your PC and back to your phone.
Jota Text Editor
Do you do a lot of writing in text files? If so, Jota is a way to do it. The interface might be old, but the program itself is solid.
DiskUsage
When your memory is all full, DiskUsage will actually chart the size your files take in a nice looking interface that sorts everything by size and can be zoomed in and out to see in details what's happening with your phone memory. Highly recommend it if you run out of memory.
Google Drive, Microsoft OneDrive, Dropbox
If you're a cloud storage user, these apps do the work of cloud usage. There's also MEGA who offers 50GBs but their problem is that they're from the same guy behind the defunct Megaupload. Won't delve into this much more.
Google Keyboard
I think its the best keyboard Android has to offer, although it lacks the thought of arrow keys to supplement it further, but the design can be set to two different design choices with different colors, and the keyboard itself is snappy and easy to use.
Smart App Lock
Might look at it if you're privacy aware. It basically locks your apps behind a lockscreen, among other things I forgot. Useful if you don't want anyone peeking into your personal documents without your explicit permission.
ScreenDim
If you want to use your phone comfortably overnights, but the screen at the lowest brightness is still bright for you, use this app to tone down the brightness even further than the phone will allow you. You can even have your phone look so dark that basically you won't be able to see anything with a light source in the room.
Greenify
It now works without root, but the more deeper functions of it still need root. However, it will help in keeping your RAM from being full by idle apps running in the background as well as save more of your battery. It will keep your device tight and in place.
As for apps for rooted phones, Titanium Backup is the very first thing you should install as a root user. It allows you to backup the apps themselves and their data, and restore them on another rooted phone. It basically holds my device tight. Lucky Patcher is a good choice for patching apps for whatever reasons you've got. Adaway is an adblocker for all of Android. Foldermount and Link2SD are good if you plan to store tons of games and your internal memory is holding you back, Foldermount being dedicated to mounting folders to your SD card and Link2SD dedicated to mounting your apps to a dedicated partition on your SD card. Oh, get Xposed Installer while you can, that's an even more powerful set of root framework and I recommend you get it, especially for Greenify.
I hope I did my best here!
EDIT: Moved Greenify from the rooted apps to the normal apps; as Greenify no longer requires root to run its basic functions but only needs it for more advanced functions.
Post edited November 09, 2015 by PookaMustard