[Aerkuld]

There is a lot of history tied up in it all but...

1. The English are British and prefer to be called English. They generally don't mind if they are called British.
2. The Scots & Welsh are British also but they don't like being refered to a British at all. They will likely become quite violent if you call them English.
3. The Irish are a little more complicated as you have;
a) Northen Ireland which is probably closest to #2, and
b) Southern Ireland which is a seperate country in it's own right and not part of Britain. They probably don't care too much what you call them as long as you're buying the round.