AppBrain Ad Detector

This app lets you see information about your installed apps: concerns, ad networks, social SDKs and developer tools. Concerns section tells you if your apps can access your contacts or accounts, may cost you money, can access sensitive data, etc.

AppBrain Ad Detector @ Google Play

Evernote

Great note manager that stores your notes online. It also has a desktop (Mac/Windows) app and a brower app. It's not perfect because you can't edit notes with complex formatting from your phone, but it usually works very well.

Evernote @ Google Play

Wunderlist 2

Nice TO-DO list manager app/widget that also lets you access your lists from a browser and stores your data online so you don't lose your TO-DO's when changing your phone. Check out Trello too.

Wunderlist 2 @ Google Play
Wunderlist 2 @ the web

Trainyard

Draw rails so the trains get to the point that matches its color. I'm not a huge fan of puzzle games but I got completely hooked on this one. I almost completed the free version and paid for more levels.

Trainyard Express (free) @ Google Play
Trainyard @ Google Play

Flash Light Widget

Simple light widget. Just turns your flash led on and off. That's it. And the icon is nice.

Flash Light Widget @ Google Play

Digital Clock Widget

After trying several clock widgets I chose this one beacuse it is simple, displays both the date and the next alarm, and lets you open your favourite alarm program when you tap on it.

Digital Clock Widget @ Google Play

Gunman Clive

Classic platform game where you jump and shoot with a cowboy. Both the design and the game play are fenomenal. I can only complain that it's a bit short, but totally worth what it costs.

Gunman Clive @ Google Play

Jet Car Stunts

3D game where you run and jump across abstract circuits that aren't very detailed but don't lack fun. I don't usually like car games but this one is different.

Jet Car Stunts @ Google Play
Demo version here

[INFO] Pirate apps can be dangerous

 
If you download a pirate app you should know that it could easily carry a keylogger. That's a program that records your keystrokes and sends it to someone else, so the hacker can see what you type and steal your messages, passwords and personal data.

Read here for more information

Open Sea!

One of the most original games I've played in my Android. Drag over the sea to open a path so the people can cross it. There are several types of characters: normal, dumb, drunk, old, mummies (the bad guys), ... and some with special powers like Mo!

Open Sea! @ Google Play

Nova Launcher Prime

This launcher lets you customize many aspects of your desktop and app drawer. I like the gestures, the app groups and how you can increase the rows and columns of your desktop and resize your widgets.

Nova Launcher Primer @ Google Play
There's a free version here

Pudding Monsters

This awesome game excels at everything a good game should: 1) Mechanics: original, simple and full of possibilities. 2) Level design: makes the most of the mechanics and keeps you interested. 3) Art design: it's a joy just to see those puddings move, blink, stick, "talk", ..

Pudding Monsters @ Google Play

Business Calendar

This is my favourite calendar. I especially like its widget and how easy it is to add/edit events and see my different Google calendars (work, personal, etc) in custom colors.

Business Calendar @ Google Play
Check out the paid version for more features.

Bobbing

Fantastic game that reminds me of a mix between Super Hexagon and Gravity Guy. If you like simple and fast paced games you must try this.

Thanks to @willeeh for this recommendation!

Bobbing @ Google Play
Free levels version here

SwiftKey keyboard

I've tried several keyboards and this one is my favourite. You can type or slide/swipe, it predicts your next word and improves with time by learning from you, and it's not necessary to select the language you want to write in.

Swiftkey keyboard @ Google Play
Trial version here

Super Stickman Golf 2

A fun minimalistic 2D golf game with powerups. Even better than the first part.

Super Stickman Golf 2 @ Google Play

JAYS Headset Control

Many headsets have 3 buttons but only one of those buttons work for Android. This app is the solution. It lets you control everything with just one button: pause, skip, change volume, etc.

EDIT: This app and others have problems with long clicks in Jelly Bean, but it's Google's fault. There's another app called Headset Button Controller that has an option to change volume by increasing it until you stop it. Not very handy, but it's the only solution (besides rooting the phone) until Google fixes the problem.

JAYS Headset Control @ Google Play