Yes, it is based on the life of Georgiana, Duchess of Devonshire, and most of the main events in the film actually did happen. The chief source for the film is the biography by Amanda Foreman, a prizewinning factual account of the Duchess' life and times.
She did indeed bring up her husband's children by another woman among her own; she was a compulsive gambler who ran up huge debts; and she was later forcibly separated from her own children. Something the movie touches on, but the book talks about in a lot more detail, is her political career. Even though women at the time had no official involvement in politics (and in fact, by no means all men had the vote either) Georgiana was very influential as a campaigner for the Whig party, led at the time by Charles James Fox. She was also interested in furthering scientific research, wrote at least one book, and was generally a more complex and intelligent woman than she appears to be in the film; but the actual facts are mostly correct.