TITLE: Why are projective objects important?
QUESTION [14 upvotes]: I belive we study them because in important categories they are close to free objects and even a retract of a free object in some algebraic instances (for example, direct summands in Mod_R, and precisely the free objects in Gps).
Am I right? Are there other reasons?

REPLY [2 votes]: If you are given a ring, it is sometimes quite non-obvious how to construct its modules. 
One natural place to look for modules is inside the ring itself and inside the free modules, which are immediately constructible from the ring itself. Now, inside free modules there can be all sort of things... but it is quite apparent that the simplest of those modules we are going to find inside free modules are precisely the summands thereof.