Usually to resign, you tip over your king. I think I read online somewhere that this is because it is less distracting to others than saying "I resign" out loud.
Offering draw and resigning

You can tip your king over if you want to... you don't have to. Generally just extend your hand and whisper "good game" or stop the clock and then extend your hand. It is not, on the other hand, proper to extend your hand when offering a draw. The reason for this is that when you make a draw offer, your opponent has until the next time that they move a piece to accept/decline. So, it's your opponent's turn and he has 1 hr and 5 mins on the clock and you offer him a draw, it's possible that he will think about it for an entire hour before deciding... an awful long time to keep your hand there ;)
I played in my first tournament recently and am still not sure what the custom is for resignations and draw offers during games. I played all of my games out to checkmate (it was my first tournament, so it was always me getting checkmated), but I don't recall hearing anybody say "I offer a draw" or "I resign," and sometimes (when watching others play), it seems they shook hands and walked away without any discussion whatsoever--I couldn't even figure out the result until I looked at the board outside the hall. So what, exactly, are the mechanics of resigning and offering draws in tournaments?