Graves are either "private"/purchased (or at least leased) or "common" aka "public", sometimes referred to as "paupers' graves".
Private graves may be used only for those the plotholder allows, e.g. members of the family, at least until the lease runs out, which tends to be anything from 25 years upwards. You generally see anything from a headstone and/or kerbs with just a few family names up to a huge monument or mausoleum. When the lease runs out and is not renewed, the plot can be resold for further burials if there's room or the remains are moved elsewhere. The original monument is usually removed and a new one can be installed, generally without the names of the original occupants, though I've seen a few where the new owner has been kind enough to include a line "Also John Smith, unrelated".
Common graves are not owned by any particular person and who is buried there is decided by the cemetery authorities. There are generally a dozen or so in each plot, though I've seen 50 (mostly children or ashes). Some cemeteries fill the grave within a few weeks to a year, so unlikely that another family member can be in the same plot, but some must work a rota and leave several years between burials. Common graves are often grouped in out-of-the way parts of the cemetery or in the middle of sections with private graves by the paths. Some places intersperse common and private graves making the area look less crowded, especially if gravestones are not allowed on the common graves. Where they are allowed, people tended to use miniature headstones, which you sometimes see one behind the other like dominos, or engraved vases or flat stones. Nowadays these tend to get moved to the edge of the plot, places around trees, or removed altogether to ease mowing easier or make cremation plots. In a few places I've seen full-sized headstones with all the names listed.
Obviously, common graves were cheaper and used by poorer families, though some who could afford a private grave preferred tot to have one, and some relatives didn't care -- the father of one well-known film star is buried in a common grave at Lambeth Cemetery because he and his son didn't get on.
Those who died without any known relatives or none who could afford the funeral were buried "on the parish" and obviously this would be in a common grave. Nowadays, it's called a "public health funeral" and usually a cremation.
Finally, there are other situations like mass graves where churchyards have been cleared or for victims of air raids or other disasters, not to mention the infamous "plague pits".
