Help me to understand.

What's the difference between Britain, Great Britain, and the United Kingdom?

Somebody explained it to me once, but it slipped through the cracks into the dusty bits of my cranium.

No difference at all - the name of the country is the 'United Kingdom of Great Britain and Northern Ireland' + lots of other bits we have scattered across the globe.

Turnip - hand in your Brit card!

Britain is a slang term for Great Britain.
Great Britain is England, Scotland and Wales.
the United Kingdom is Great Britain and Northern Ireland.

I'm not too sure where the outlying islands (Isle of Man, Channel islands, Hebredies, etc.) fit into it all though.

I believe the islands come under the UK,  even Gibralter, but I'm not sure if the Falklands and the other islands further afield are part of the UK or just belong to the UK, but then who cares, unless some little tin-pot, banana dictator wants to annex them?

Well, that makes it crystal clear.

I had some commentary from a couple Brits on the way things are organized (?) over here.

We have separate police departments at the town/city, county, and state levels, in addition to federal law enforcement agencies such as the FBI and US Marshalls.  Also, each state has its own drivers license.  It doesn't make for a homogeneous system.

Basically, we have 50 independent political entities joined together in a federal union.

The key is that the states are supposed to retain sovereignty over their internal matters in all areas where the federal government does not have specific Constitutional authority.  Unfortunately, since our Civil War the federal government has continually expanded and encroached into areas not allowed by the Constitution.

Quoted: Well, that makes it crystal clear. I had some commentary from a couple Brits on the way things are organized (?) over here. We have separate police departments at the town/city, county, and state levels, in addition to federal law enforcement agencies such as the FBI and US Marshalls.  Also, each state has its own drivers license.  It doesn't make for a homogeneous system. Basically, we have 50 independent political entities joined together in a federal union. The key is that the states are supposed to retain sovereignty over their internal matters in all areas where the federal government does not have specific Constitutional authority.  Unfortunately, since our Civil War the federal government has continually expanded and encroached into areas not allowed by the Constitution.

I think it really got bad during and after the "New Deal" era. The depression made people desperate and they bought into FDR's programs. But I think there is now a trend back towards State's rights, it will never be what it should've been under the constitution. In a country as large as ours, state's rights is the only practical way to go. And it's more democratic. Governors and State legislators know and care more about their state than do distant Washington officials.

Quoted: Basically, we have 50 independent political entities joined together in a federal union.

Much like the plans for Europe.