In Pokemon GO, you can often find Eggs in the wild, usually by interacting with PokeStops to find free items
There are different kinds of Pokemon Eggs which can hatch into numerous Pokemon, but to hatch them you'll have to do a bit of work. Each player starts with one Egg Incubator, which allows you to hatch an egg after a certain amount of steps are taken, tracked by your phone's GPS and the game.
To hatch an egg, check your Pokemon list and hit the tab labled "Eggs". Clicking on the Egg will prompt you to put it in an incubator to begin the process of hatching said Egg. Note that you only have one incubator to start with, but more can be bought from the Shop, although the shop versions will break after 3 uses - while your starting incubator has unlimited uses.
Once in the incubator, you'll be given an indication of how long it will take (usually in kilometers) to travel in order to hatch the egg. Then its just a matter of going off to explore while your GPS does the work and the egg will soon hatch!