She plays Dr. Meredith Grey, a young doctor who joins the staff of a Seattle hospital as a surgical intern, eventually becoming a resident, then an attending, surgeon. Her life is complicated by the fact that her father is Ellis Grey, a famous surgeon who suffers from Alzheimer's disease, and by a one-night stand she had with a man just before she joined the staff of what is now Grey Sloan Memorial Hospital (at first called Seattle Grace) who turns out to be an attending neurosurgeon at the hospital. The surgeon, Derek Shepherd (played by Patrick Dempsey), pursues Dr. Grey romantically; she eventually relents and marries him. All the while, Dr. Grey deals with her work and motherhood and gets involved with her friends and colleagues.
All heady stuff, and the show is, as of this writing, still going strong.
Ellen Pompeo had also played one-shot character on TV shows such as "Law and Order" and "Friends," and she's had roles in movies such as Catch Me If You Can and Eternal Sunshine of the Spotless Mind.