Most people would answer: In a nest, of course! But it is a tad more complicated than that. For instance, there are some mammals in Australia and New Guinea that produce milk to nourish their young, but they lay eggs and care for them until they hatch. The duck-billed platypus lays its eggs in a burrow, where the two eggs stick to each other and to the female’s fur while the female incubates with her tail wrapped around the eggs. In a sense, the female herself is the nest, and the burrow houses the female. In contrast, a spiny echidna puts her single egg into a pouch, like that in which marsupials (e.g., kangaroos) keep their offspring, until it hatches. So echidna females carry their eggs with them as they forage.
But birds typically don’t carry their eggs around; they put them in special places. It would be easy to guess that birds don’t lug their eggs around because they are basically flyers (although some have reverted to walking) and it could be very awkward to tote eggs while flying. But sungrebes in Latin America are able to fly while carrying chicks in pockets under their wings, so why not eggs too? Furthermore, even though they did not fly, lots of dinosaurs made nests: fossil eggs have been found in ancient nests, sometimes with a presumed parental dinosaur fossilized nearby. That leaves open the question of why birds usually have nests for eggs.
One possibility is that it would be hard to fly with more than about two eggs in some pocket and having a nest would allow the production of larger clutches of eggs. Some birds lay large clutches of six or eight eggs, or even more. Having more offspring could mean that at least some might survive to adulthood and produce the next generation.
Bird nests come in a huge array of forms. Some birds, such as most shorebirds, just make a little scrape in the soil or sand, sometimes adding bits of stone or grass. Others build substantial piles of vegetation in marshes or in trees. Some weave complex chambers or hanging bags, some make a sketchy latticework of sticks, and others build cup-shaped nests of various fibers. Some birds build nests of mud, shaped in various ways, and even of their own saliva (think: bird nest soup!). Some birds nest in cavities made by other creatures, and others carve their own cavity from a tree or earthen bank. And all that just begins to indicate the variety of nests used by birds.

Even among our local birds, the variety of nests is impressive. Some examples: Barn swallows stick a little ledge of mud pellets to a wall and line it with feathers. Cliff swallows take it one step farther and build a gourd-shaped mud nest with a narrow entrance. Juncos make a grass-lined cup on the ground, while hermit thrushes make a mossy nest in a shrub or small tree. Killdeer and spotted sandpipers carve out shallow saucers in the sand. Ducks generally make downy bowls on the ground, but some have taken to tree cavities. Woodpeckers chisel their own nesting cavities in trees, but tree swallows, nuthatches, and chickadees depend on old woodpecker holes for their nests. Solitary sandpipers take over abandoned robin nests and great horned owls often use old nests of red-tailed hawks.

Although some kind of nest is the norm for breeding birds, there are some oddball birds that don’t make nests. For example, emperor penguins put the single egg on the feet of the incubating male, and a fold of feathery belly skin drapes down to cover the egg; the male can actually shuffle around on the ice while holding his egg safely on his feet. Fairy terns in the South Pacific perch their single eggs on a bare branch, somehow keeping them from falling off. Cliff-nesting seabirds often just balance their eggs on a ledge.
Once the eggs are in a basket, so to speak, there is an inevitable constraint on the behavior of the parents. They (or at least one of them) are now tied to a specific place for a certain period of time while the eggs are incubated and, in many cases, while the chicks are growing up. Being tied to a central place that is the focus of activity gives human nest-searchers many clues about finding scattered nests; we commonly just try to follow songbird adults as they carry nesting material or food to the central place. Nest predators are likely to do the same thing: jays and ravens and crows are strongly suspected to do so, for example, and weasels and foxes surely can do the same.
A few birds avoid the central-place focus by dumping eggs in the nests of other birds, often other species. American cowbirds, Eurasian cuckoos, African indigobirds and honeyguides, and some ducks, for example, have become brood parasites, letting other birds rear their chicks. The foster parents still have the central-place focus, of course, but the brood parasites have dispersed their eggs among several foster parents and at least diffused the potential problem.
When parent birds are central-place foragers during incubation and chick rearing, the nest needs to be located within range of a sufficient food supply. Few songbirds fly more than a few hundred yards from the nest to find food, although American Dippers sometimes fly a mile or more along a stream. An extreme contrast is seen in some seabirds, such as albatrosses. For example, the Waved Albatross of the Galapagos Islands may fly more than sixty miles from its nesting site to find food; other species may fly even farther. The interval between chick-feeding trips may be several days, for some kinds of shearwaters, because the parents are ranging so widely over the ocean in search of food.