GAME TROPHIES - can only be awarded after a game has ended. There's a Give a Trophy link next to the board.
FUN TROPHIES can be given at any time. Either click on a member's name then notice the Give a Trophy link on the right-hand side (under Actions).
Or hover over a member's name in a forum posting, a hover menu appears which includes the option: Give a Trophy.
How do I send trophies?