To reserve a gift, open the wishlist link, click Reserve on the wish you plan to give, and confirm. This marks the item as taken so no one else buys the same thing — but it does not buy the item or place an order. You still buy it yourself in the external store afterwards.
Steps
Open the wishlist link the owner shared with you.
Find the wish you plan to give.
Click Reserve on that wish.
Confirm the reservation.
That's it — the item now shows as reserved, and a second person can't reserve it until you cancel.
A reservation is only a signal that says "I've got this one," so people don't double-buy; you still have to buy the item yourself, as described in "How do I buy a gift from someone's wishlist?" And because each item can be reserved by only one person at a time, trying to reserve something already taken will tell you that someone else has it. Reserving doesn't complete the wish either — after you buy, only the owner can mark the item as received. See "What should I do after buying the gift?"
Common questions
Do you need an account? That depends on the owner's settings. Some wishlists let anyone reserve without signing in; others require you to sign in first; and on some, reserving is turned off entirely.
Reserved without an account, then signed up? If you stay in the same browser session when you create your account, your guest reservations carry over to it automatically. Reservations you made in a different browser or on another device won't be linked.
Changed your mind? You can cancel your own reservation, which frees the item for someone else. See "How to cancel a reservation" in Gift Reservation.
By default the owner doesn't see that anything was reserved, and your name stays hidden unless their settings show it — so the surprise stays safe. See "Can I see who reserved or bought a gift?" in Gift Reservation.