Or is the body buried elsewhere and the slab is merely a memorial?
It can vary from church to church. Somtimes it's a plaque and the person is buried outside, sometimes they're under the floor, sometimes they're under the floor vertically (I'd love to know how they went about those burials), sometimes it's an urn with the ashes, and so on and so forth.
If you look for example at Westmister Abbey, you can see there's just about every variation on the theme:
If you have the time, Historical memorials of Westminster Abbey goes into detail and includes sketches of some of the vaults in the church. A simplified answer is that there is a subfloor below in which stone slabs set upright support the edges of the stone floor. Only those who could afford to pay were buried in a church, the rest were buried in the grave yard. But being buried in a church didn't necessarily protect you from being dug up and someone else put in your place. Shakespeare's wife was displaced in the 18th century in the Stratford church and another person buried in her place. Her stone was lost, and in the 19th century a new brass marker was cut using William Dugdale's transcription published in his Antiquities of Warwickshire (1656).
EDIT: It read as if I were discussing one church; ce to eliminate confusion.
EDIT: Be sure to read this account of the search for the grave of James I in the appendix.