Hey.. I made an animation in after effects a while ago :
as you can see it is in the bottom of a 1920x1080 video. I made it that way so i can see better the size of it on a screen this big.
As rendering goes i really want to render it in 500x500 to save space and time, and also with transparency,if possible,so i can put it over any video easyly.
How can i do that ?
Not sure what you are asking. The size of the comp is irrelevant for the render speed if it only contains nothing. Everything else is a matter of resizing/ cropping the comp in the comp settings and chosing a suitable output format that supports transparency, on all of which there's plenty of information here in this forum, the AE help and external tutorials.
I guess that you mean that you want to crop all the empty space and keep the 500x500 only for your logo at the original scale that would have in the 1080p video.
I'd recommend first of all precomposing the scene for keeping the original one untouchable, and then putting the composition settings 500x500 and moving the precomposition until your logo is in the middle of the screen.