Briton is the most widely accepted term for people from Britain (which, of course, is not the same as England and the United Kingdom). Britisher had a brief heyday in the 20th century, but it was always only an American term and was never accepted by Britons themselves. Brit is not offensive, but it is informal.
Of these words, Brit appears most often because it serves as both a noun and an adjective. Briton is only a noun, and it only denotes people. Britain is always the correct spelling of the place name.
Note: We’re American. Please let us know if we’re missing anything.
Examples
For example, these major British publications use Briton:
Sir Howard Stringer is the Briton who chose to serve in Vietnam so he could stay in the United States … [Guardian]
In 2000, he became the first Briton to win the Hans Christian Andersen Illustrator’s Award for services to children’s literature. [Financial Times]
He is also understood to be the youngest Briton to stand atop the Himalayan summit. [Evening Standard]
And these non-British publications likewise prefer the widely accepted term over the alternatives:
An expatriate Briton, Hadley was a Roman Catholic convert who devoted his life to society’s casualties. [Montreal Gazette]
The 68-year-old Briton used to get so nervous before shoots that he’d vomit. [Forbes]
“No he wasn’t,” the Briton scoffed. [Stuff.co.nz]

