When I redeem a game code, the expiration date is X, is it the expiration date of the key or the expiration date of when the game will be removed from my account?
This question applies to both game codes and game gifts.

no.
Once redeemed it's yours forever.

If the code is issued by GOG, than the expiration date refers to the date when the key can no longer be redeemed. Typically, GOG does not remove games from a user's account unless the game was a.) purchased using stolen credit card information, or b.) the user requested the game be removed.

EDIT: ninja'd by less than 60 seconds!

Expiration date for the code. Usually only applies to "free" codes you get from events and sales. Purchased games/codes never expire, even if game is removed from the store (you still keep the game and as far as I know, can still redeem it). I have some from years ago that are still redeemable.

Thanks guys :)

In addition to what everyone else said above, if you have a game code in your order history that is not yet redeemed and you either own it or aren't personally interested in it (although that would normally only happen if you already own it in your account and it is a spare code), if it is nearing expiration and you don't have a friend or someone to gift it to before it expires, you may consider contributing the code to one of the forum "Community Giveaway" threads by sending a private message to the giveaway co-ordinator, or alternatively holding your own forum giveaway or similar. If doing so be sure to make the giveaway end with a day or more to spare for the winner to be able to receive and redeem the code. :)

Just make sure your code avoids bad influences until it becomes a good member of the community again.