A nursery school or daycare was an educational establishment for very young children who were not ready for school.
Vulcans had nursery schools. One of the activities the children there engaged in was dance, some of which resembled Marta's Orion dancing. In 2268, experiencing a moment of nostalgia, Spock compared their dances to hers, but noted that she had better coordination than the children. (TOS: "Whom Gods Destroy")
Deep Space 9 had a day care. In 2370, "Miles O'Brien" offered to drop Molly off at day care 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")