A user's Collection is viewable from My Geek.
Note that your "Collection" does not necessarily mean games you own in real life. It is a virtual collection of games you are interested in due to owning them, playing them, rating them, commenting on them, wanting to be notified of new content, whatever.
To add a game to your collection, see "How Do I Add a Game to My Collection?" in BoardGameGeek FAQ.
You can add several copies of the same game to your collection, e.g. because you own several copies, or because you have played several different versions and want to rate/comment them individually. If you rate a game multiple times in this way, then only your highest rating for the game is used.