The full name of our country is:
The United Kingdom of Great Britain and Northern Ireland. The Great Britain part is made up of the ancient kingdoms of England and Scotland, and the principality of Wales. So, our country is in fact made up of 4 countries. In some sports, football, rugby, cricket etc. we play as individual countries. But sometimes, we play as the British lions, normally in some rugby matchs, this is all 4 countries making up 1 team.
Then again, in other sports we compete as 1 team, Great Britain, but never the United Kingdom.
As for the flag. The Union Jack, or Union Flag, is the flag of the UK. It is made up by the individual flags of England, Scotland, Wales and Ireland. It is the crosses of St George (England), St Andrew (Scotland), St David (Wales) and St Patrick (Ireland).
Hope this helps, we even get confused sometimes!
Over here in the UK, we drive on the left, so the steering wheel is on the right.