A daycare was a place where young children who were not ready for school yet were sent.
Deep Space 9 had a daycare. In 2370, "Miles O'Brien" offered to drop Molly off at daycare on his way to ops, but Keiko O'Brien said she would take her to school instead. When "Miles" asked why, Keiko explained that she wanted to show Molly a Vulcan problem solving educational program. In reality, Keiko was trying to keep Molly away from a replicant of Miles O'Brien. (DS9: "Whispers")