How to gift

Add friend on Steam, then send it to them.

sent it via steam or sent the key from chat.

add to friendlist and send it via steam

...and just to be extra clear if gifting is new and confusing:
You must already have a gift copy of the game in your Steam inventory, or buy a copy of the game from the Steam store and send to the winner (after adding them as a friend). You can't remove a game from your Steam Library (the games on your account that you can play) and send that copy of the game as a gift to someone else. It must be a separate gift purchase.

...and of course, if you have (or buy) an unused key for the game, you can always send that to the winner, even if you chose "gift" when you first created the giveaway.