Britton was born Constance Womack in Boston, Massachusetts, the daughter of Linda and Allen Womack, who was a physicist. When she was seven years old, her family (mother Linda, father Allen, and sister Cynthia) moved to Lynchburg, Virginia. She majored in Asian Studies at Dartmouth College (class of 1989). Upon graduation she moved to New York City, where she spent two years at the Neighborhood Playhouse studying with Sanford Meisner. Britton uses her married name as her stage name, though she is now divorced from her husband.

While studying at the Neighborhood Playhouse, Britton (then Womack) made her NY theatrical debut in Caroline Kava's The Early Girl at The Courtyard Playhouse. Britton played seasoned prostitute, Laurel, opposite then unknown television and radio personality Cooper Lawrence, who played Joan. Britton's performance, while well received, nearly got her ousted from the Neighborhood Playhouse program, which prohibited students from taking professional employment during their course of study. After graduating, Britton spent two more years working in off-Broadway theatre productions. She moved to Los Angeles after the success of "The Brothers McMullen". In her free time, Britton, who resides in both New York and Los Angeles, enjoys hiking, yoga, and doing volunteer work.

Britton had a recurring role on 24, during its fifth season as Diane Huxley, the landlady of protagonist Jack Bauer (Kiefer Sutherland. She co-starred in the indie-thriller The Last Winter in 2006.