Legally, pre-school and playgroup and nursery are the same thing, each place just chooses whether they are registered for sessional or day care. In fact where I live several playgroups have renamed themselves nurseries because it sounds posher and more like private ones!
But they all follow exactly the same curriculum and are subject to the same Ofsted and Social Services inspections. (Same as reception classes in schools in fact)
The real difference is that private nurseries and the government ones like Surestart usually open around 8-6 and often do holiday periods too, whereas non-profit-making pre-schools, playgroups and nurseries only operate for 39 weeks a year. Private nurseries charge much more too.
The ones with longer hours are often targeted at working parents. The non-profit ones offer 'sessional care', with sessions lasting about 2 1/2 hours, although at many you can opt to pay a little extra for a lunchtime session of 45 mins or so to bridge the morning and afternoon sessions, thus giving you the equivalent of a full school day.
At a non-profit one, your child will be totally funded by the government from the start of the term following their 3rd birthday - at a private nursery you would have to 'top-up' to meet the fees. Before they get government funding, non-profit ones usually charge modest fees - ours is £6 a session, compared to about double that at a private nursery.
There is some overlap - for example, a private nursery near me only opens 9-3, so similar to many playgroups, and children all have to take packed lunches. Other nurseries provide cooked lunches and are open longer hours.
Nursery schools are different again. They are a pre-school unit attached to a primary school. Admissions vary, but they will generally accept children from about 3 or 3 1/2. Often at schools with nursery units, the age at which children start 'school proper' is later - where I lived in Berkshire, my child went to a school nursery from 3 1/2 and would have started school in the term before she was 5. However, in the county where I live now, there are almost no school nurseries, and children stay at playgroup/nursery before starting school in the Sept following their 4th birthday - only one intake.
Groups where parents stay with their children and play for a couple of hours are called toddler groups (used to be Mums & Tots but many Dads now too!)
Confused?