Yeah I know that thousand of sites and people already write one of this lists. So why Im writing this? Because I see thousand of pages talking about thousand of apps not so necessary, so I made my own. Let's go
The "must have" backup software for android. You MUST have root access to use it, cause he can make backup to system applications (with configurations), very useful if you like testing some roms. There's a free version and a paid version with nice features.
There's loads of apps that do this job. You can track your cellphone, get info, last calls, remote lock, remote wipe (internal and sdcard) and alarm. The differrence from the others is that one only costs 0,99 USD (one time only, and it works).
Camera360 / Camera360 Ultimate
A nice camera application with lots of nice features, such effects, plugins, etc. Isn't VERY NECESSARY, but if you like to take some shots, give a try. There's a free and a paid version with some features.
An automated task scheduler, with him you can, for example: "I want my GPS only working with applications X and Y", "I want my wifi on only when Im at home", and so on. Of course, its an advanced application, you need some time to learn how it works but its perfect. There's no free version.
If you are addicted to forums (like me), this is the best you can have. You can test the "feel" using the XDA-Developers Forum application.
Its the same widget buttons app from native android, but this one you can customize with thousands of options.