You are correct - when it refers to hand it means the cards you didn't use in your word. And yes it allows you to trash a card, you don't have to wait for another card.
So in order to use a card that gives you a power when you trash a card you don't have to have also played a card that says "trash this card" to benifit from the card that said you get a power if you trash a card. Ok Thanks for the clarification as I had the same question, but to add to it...
In order to trash a card from your hand, do you have had to play a card that mentions trashing? Or other words can you just choose to trash on your own because you want to cull your deck?