Warning. Brief history lesson ahead.
The Regency in British history occurred when the Prince Regent took over for his um, less than sane dad, King George III, who was mentally incapacitated and unable to rule properly. That's why we call this period the Regency, because the Prince Regent was in charge, not the king. The formal Regency lasted from 1811 to 1820.
The Regency period in literature is a broader period of time and dates between 1790 and 1830, beginning just after the French Revolution. While many of the Romantic poets were active at this time, this era is probably most remembered for the domestic novels that we all know and love.
Jane Austen is one of the most famous novelists of the Regency period, and when you read her works, it will come as no surprise that the Regency was all about elegance, refinement, manners, and class. Get out your white gloves and lace ribbons. We're gettin' fancy.
Of course, like most important eras in English literature, the Regency period had a lasting impact on the future. James Makepeace Thackeray's Victorian novel Vanity Fair is set in the Regency. So is Tom Stoppard's play Arcadia (to be fair, that one's also set in 1989, but we'll just have to let that part slide).
For some reason, contemporary peeps are also way obsessed with the Regency. Maybe it's the fashion. Or maybe it's just Mr. Darcy. Sigh.