[pau1f0rd]

As Ireland is part of the British Isles, surely the Irish are British too?

[yarddog]

It is sort of hard to keep up with, but I think Great Britain is composed of "England, Scotland, and Wales".

The United Kingdom adds Northern Ireland. Of course many Americans simply call them all "English" sort of like we called all members of the Soviet Union, "Russians".

[tjwmason]

As Ireland is part of the British Isles, surely the Irish are British too?

The Irish question is highly vexed, as I am sure you are aware. Before 1922, undoubtably the Irish could have been counted as British, however at that time most of Ireland left the United Kingdom (to form the Irish Free State, later the Irish Republic), and did so with little affection for us; calling a southern Irishman British would generally be somewhat of a faux pas. Taking a purely technical definition, a British Citizen is a citizen of the U.K. (England, Scotland, Wales and Northern Ireland); this is probably the best one to use, though one should still be careful with the Northern Irish, 60% would love you for calling them British and 40% would hate you.