Prerequisite-Rooted Android Device and working internet connection.
Most of the Apps listed here are free, some have pro versions which require you to shell out some cash, but don't worry google's to the rescue. You can find their apk's and get the pro version for free, just google it.
1.DroidVPN- It let's you access sites normally blocked in colleges or in offices. It requires you to register with them, but then the registration is free. You can access any website without any limitations. Here
2.BusyBox- It's for fans of linux commands it would make your android run commands like telnet and ping but prior to that you'll require a terminal emulator listed here. Also this app is a prerequisite while installing many apps that tweak your device. Here If you don't want to pay for it, then google it, you'll find it's apk.
3.Terminal Emulator- For linux fans who like to try linux commands in their android. It requires a BusyBox to run with full capability. Here
4.DosBox turbo- If your a fan of old pc games and would like them to run on your android than this is the app you're looking for this is an x86 emulator. It can also make Turbo C++ run on your android.
**Read instructions on how to use carefully, otherwise it'll be a headache for you.
type intro then type intro mount you'll get the list of instructions follow them carefully.Here It also has a pro version but the free version is as much better.
5.Hackers Keyboard-If your using DosBox turbo or any other x86 emulator than you'll require this keyboard, this keyboard has all the keys like control key, function keys. It brings the real old pc keyboard on your android.
6.YouTube downloader- For people who like to watch you tube videos by downloading them. It downloads videos as you surf through youtube, just click on share and choose YouTube downloader and the rest it'll do.
It's not available on the playstore, It can be downloaded from here.
7.RootBrowser- It can browse through the system of your android also giving you the capability to modify system files, use with care and don't mess with files you don't know about.
8.XposedInstaller- For people who don't want to give up stock rom but would still like some changes in it, than this is the app your looking for. Again not available on playstore, it's available on xda site.
9.Gravity Box- Additional changes in rom can be made by installing this app, this app requires xposed installer. You can add power button tweaks, status bar tweaks, notification bar tweaks to name a few. Must have Xposed installer to make it work. Can be downloaded from the downloads option of Xposed Installer.
10.Lucky Patcher- It can remove the annoying ads from many apps, also patch apps in many ways, like removing license verification and so on. Useful to users who usually don't download apps from the playstore. It's not available on playstore, here is the official website's link.
11.BootAnimations-It can change the boring default boot animation of your android. You are given a list of animations to choose from. Also it contains bootanimations for Tablets as well. This one on the left is what i've chose as bootanimation. Here
12.Script Manager- Handy app which comes in use while applying tweaks for the android by running various scripts, better to have this app if you constantly like tweaking your device.
If you want to test it download V6 supercharger script from xda. It speeds up the device by making many changes in the system files and requires script manager to run it. Here
12.DroidCam- It's the app you should have if you're not having a webcam. It will turn the back or the front camera of your phone into a webcam. It allows you to have a wireless camera via wifi or you can connect with a usb. It's pro version supports connectivity over bluetooth. Here
13.BugTroid- The ultimate app for a hacker, contains all the available apps for hacking, be it wifi hacking, sniffing, Ddos, pentesting. It's all available in this app.
I think you know what to do if you don't wanna pay for it.
The hint is "Google it".
14.Orbot- The equivalent of tor for android. Let's you browse the web anonymously. Works better with Orweb v2.
15.Barcode Scanner- Probably the best scanner. It scans the QR codes very fast. Here
*Root not required
16.Real Chess Clock- As it's name suggests, a clock for playing timed chess matches. Has a very good interface. Here
*Root not required
***Almost all the Apps Listed above are free and can be downloaded from the Google Playstore.
Xposed installer and Youtube downloader can be downloaded from the xda site, you'll find it's link there.
The apps listed above are a must if you would like to bring more out of your rooted android device.
Check out xda site, it has tons of apps for rooted devices.