To adopt a child, you first must own a home with a child's room. This can either be done by purchasing a home in a city and then renovating a child's room after talking with the steward, or by adding a child's wing to your custom house.
There are two ways of finding a valid child for adoption. The easiest way is to find one of the many homeless children in the cities, give them gold (charity) and then choosing the unique dialogue afterwards. The other way is to adopt from the Honorhall Orphanage in Riften, though this requires you to have killed Grelod the Kind during the 'Innocence Lost' quest, and having received a letter some time after from Constance Michel.