Which birds do not have wings?

All birds have wings. It's just that some of them are useless for flight.
Even the New Zealand kiwi has wings, though it appears to have none. The wings are small and rudimentary, hidden under the kiwis' hairy feathers, but certainly present.

Some of the flightless birds are members of the ratite family. Other ratites besides the kiwi include the emu of Australia, Southern cassowary of Australia and New Guinea, the ostrich from Africa (the largest of the flightless birds) and the rhea from South America. Ratites have wings but the bones in their chests do not have the capacity for flight muscles, which is what a bird needs to fly.

Penguins cannot fly, using their wings instead as flippers, but they are not ratites. Nor is the kakapo of New Zealand, the world's only flightless parrot. Tinamous are ground-dwelling, flightless birds (about 47 species) which are related to ratites; they live in Central and South America.