The simple answer to that question is "I'm in all three." England, Scotland, Wales and Northern Ireland are more like "super-autonomous regions" than they are true, individual countries. The correct term is "constituent countries," and they fall somewhere in between province and sovereign nation on the spectrum (but closer to the latter). They have their own ways of practicing law, different taxes, and on and on. They also have their own parliaments, senses of national pride, flags and cultures. In 1998, important acts of Parliament granted Wales, Scotland and Northern Ireland much more independence than they had in the past, but the four constituent countries still make up one truly sovereign nation.
Without getting too much into history, the four had become united over time to form the United Kingdom of Great Britain and Northern Ireland. That long name is the official name of the single, sovereign country. Queen Elizabeth II and the Royal Family reign over the entire UK. "Great Britain" simply refers to the island that encompasses England, Scotland and Wales. However, Northern Ireland and multiple small islands are theoretically included by that cumbersome seven-word name. So when somebody says "I'm British," it's basically the easiest way of saying "I'm United Kingdom of Great Britian and Northern Ireland-ish." If they say "I'm Welsh" or "I'm Scottish," then they're simply telling you what part of the UK they're from.
One of the neat byproducts of the union of these four countries is the UK flag. You're all probably familiar with it, but there is a good chance that you didn't realize (because I didn't until last winter when it hit me like a Mack truck) that it's just a clever combination of the English, Scottish and Northern Irish flags:
Here's the Wikipedia entry in case you're still curious or confused. However, those who are very proud of their Irish Catholic heritage should be warned. The history of the UK is not particularly rosy in that regard.