Open the list on Wishpicks, tap the wish you'll buy, then tap Reserve and confirm; other gifters see it's taken, so nobody buys the same gift.
Steps
On a phone:
Open the list's link and tap the wish you want to give.
Tap Reserve in the bar at the bottom of the screen.
Tap Reserve again in Confirm reservation.
On a computer:
Open the list and hover over the wish's card.
Click Reserve on the card, or open the wish and click Reserve there.
Click Reserve in Confirm reservation.
A Reserved window then opens, with Buy on [shop] if the wish links to a shop and Cancel reservation if you picked the wrong one. Want to leave the owner a note? Tap Leave a message for the owner before you close it: see "Will they know it was me who reserved it?". You pay in the shop: see "Can I buy a gift on Wishpicks?".
Faster on a phone: Reserve at the top of a card's ⋯ menu reserves at once, with no confirm step, on lists that don't ask you to sign in.
You can also reserve straight from Home, under From people you follow.
Why can't I reserve on this list?
Usually the wish is taken, the list asks you to sign in, you help run it, or reserving is off.
Someone else has reserved it, or it's marked Fulfilled: tap Only what's left to see what's still free.
You see Sign in to reserve: sign in first, then reserve.
It's your own list, or you were invited to help run it: see "How do reservations work?" for who can reserve.
The wish says the owner "has turned off reserving on this list": ask them to turn it back on.
Why does this list ask me to sign in?
The list's owner chose to take reservations only from signed-in people, so your reservation carries your name.
Tap Sign in to reserve.
Tap Sign in, then sign in or sign up (see "How do I sign up or sign in?").
You're back on the wish: tap Reserve and confirm. Worried about the name? See "Will they know it was me who reserved it?".
How do I see only the gifts nobody has reserved?
Tap Only what's left at the top of the list. The header shows how many are left, and if none are, you'll see "Everything here is taken".
Can I add something from their list to my own?
Yes: open the ⋯ menu on the wish's card and choose Add to my wishlist. It copies the wish to a list of yours and leaves theirs as it is.
Tips
Tap Top priority at the top of the list to see what the owner wants most.
Reserve as soon as you've decided, even if you'll buy it later, so nobody else picks it while you wait.
Buying with a friend? Send them the wish with Share in its ⋯ menu, and see "Can I buy a gift on Wishpicks?" for sharing the cost.
Reserving without an account? Read "Where can I see the gifts I've reserved?" before you do, so you can find the gift again later.