Geographically they are all part of the British Isles.
Historically / Politically:
Great Britain = England (including Wales) + Scotland
United Kingdom = England (including Wales) + Scotland + Ireland (now only Northern Ireland)... this is similar to the United States (which is a collection of states).
The British are the citizens of the British Isles / UK. But I think they identify themselves as English (Anglo-Saxon), Scotish, Welsh, Irish (the latter three are Celtic I think)...
The flag of UK is a combination of the English (St. George's Cross), Scotish (St. Andrew's Cross), and Irish flags (St. Patrick's Cross). Sorry Welsh simbols aren't included.
I think that difference between the UK and the US is that the UK has a rich history (pre-history, ancient, medieval) compared to the US (not counting native Americans)... where there were several established kingdoms and are indigenous inhabitants of the lands (maybe not the Anglo-saxons). So the UK is truely a collection of nations. As Punk mentioned, the Celtic kingdoms are more individually nationalistic (calling themselves Welsh, Scottish, Irish) while the English are more collectively nationalistic (calling themselves British). On the other hand, majority of the US populaiton are immigrants (Anglo-saxons again(?)... I see a pattern here.) and the States are just political boundaries instead of true nations. There is not much inclination for individual state nationalism but only collective nationalism since they are racially homogenous (not these days) and without much historical attachment to the land during the establishment of the US. By the way, didn't the Native Americans consider themselves as individual "nations"?
To the Brits and the Americans, am I correct to draw these conclusions?
In answer to ilovestrategy I would say it was not a dumb question - you'd be surprised how many people from the UK do not understand....no seriously
The British Isles is made up of three Kingdoms (England, Scotland, Ireland) and one principality (Wales). That is the reason the Welsh flag does not appear on the flag of the United Kingdom.
As can be imagined with countries so close together, war was a regular feature between these peoples both before, during and after the Romans left.
The English invaded Wales pretty early in preceedings (1300's?) and I guess sort of "annexed" the country. English and Welsh tend to share the same laws to this day as a result of this "annexation" for example.
Things get really complicated with the end of the Tudors (1603) and the English king James I also being James VI of Scotland (due to Henry VII's sisters marrying into the Scottish royal family and Queen Elizabeth dying childless). But the formal act of Union between England and Scotland which united the two Kingdoms, only came about in 1707.
At this time Scotland - as an established Kingdom in her own right - kept many of her own laws and does to this day). At this time the Kingdom's flag was the cross of St George (England) and the Saltire (St Andrews cross of Scotland). England Scotland and Wales together form Great Britain.
Ireland was invaded at various times and there's some great (and sad) history surrounding at various times, Oliver Cromwell, the Pope and William of Orange, James II etc.
A formal act of union took place in 1801 joing Ireland with Scotland and England under the name, the United Kingdom of Great Britain and Ireland. The red cross of St Patrick was added at this time to form the flag we know today.
I won't begin to go into the tortuous Republic of Ireland / Northern Ireland split, suffice to say this happened in 1921(?) and the Republic became independent of the UK. Northern Ireland remained ruled from London (Westminster).
The term British therefore does not technically include Northern Ireland but this is generally ignored - its easier on the tongue than United Kingdom.
In recent years there have been moves to devolve power from Parliament at Westminster to local parliaments in Scotland, Wales and (again I won't go into Ireland).
So as you can see, we are not like the USA - you are one country, while we are four - and one at the same time.
Personally, I was brought up by my parents (both of whom served their country in WWII) to be British and proud of it. In recent years I have felt myself becoming more drawn toward my Englishness in response to a lot of anti-English political nonsense at home (that I won't bore you with). But despite this, I still love the UK, its rich and varied history and the contribution of all her peoples English, Scottish, Irish and Welsh, to extending civilisation and democracy across the globe. I still firmly believe she is a country greater than the sum of her parts.
England expects that every man will do his duty. Horatio Nelson October 1805