Why do some apps require a password when I move them to the trash? Some applications located in /Applications require a password in order to move them to the trash, others don't. Then there are also apps which cannot be deleted because they are required by OS X (a warning pops up).

Upon further inspection, I've come to the conclusion that Apps installed via the Mac App Store require a password in order to be moved to the trash. Why?
This is a permissions effect.  Most Application installed on Lion by default, via the App Store, and some installers are installed as system a.k.a Root and root only by default has Read and Write access to the Applications installed as root, while everyone else including your user is read only or custom for each Application. Applications that you drag and drop are owned by you, and you can delete them at will usually with out a password.

Verify in the Finder or Terminal what the permissions are on Apps that require a password to move to the trash, its likely that those Apps that are set to read only access for your current user are the ones asking for a password and a temporary privilege to delete something owned by the system. 

In terminal try:

ls -la /Applications