The short answer is : the difference is not as big as you think. People speaking standard British English and people speaking standard American English can always understand each other, and so can anybody else. Dialects is a different matter. People from New York sometimes have trouble understanding farmers from Texas, and people from London sometimes have trouble understanding taxi-drivers in Glasgow in Scotland. And obviously, the New Yorkers would find it even harder to understand the Glasgow taxi-drivers, just as the Londoners might struggle more than than New Yorkers to understand the Texan farmers. Some slang and colloquialisms are different, but slang and colloquialisms vary even within countries. The other thing to remember is that British and American written language is almost identical, give or take a few very minor spelling differences. If you want to know some precise details, this page is pretty comprehensive: http://en.wikipedia.org/wiki/Comparison_of_American_and_British_English

Mostly it's different slang and idioms. The only significant difference is the spelling of some words (e.g., "color" vs. "colour") and some different prepositions (e.g., "on the weekend" vs. "at the weekend"). There are also some pronunciation differences that go beyond accent (e.g., "aluminum").