How do I redeem a Steam gift card code?

In the Steam client: Games menu at the top, then Redeem a Steam Wallet Code, and paste it in. On the store side it's under your account details, Add funds to your Steam Wallet, and there's a redeem link on that page. The thing that trips people up: "Activate a Product on Steam" is right next to it in the same menu, but that box is for game keys and will tell you a gift card code is invalid.