A lot of the interpersonal relationships have been developing and changing since the beginning of the show, but it's not like "Desperate Housewives" where you need to have been watching it to understand what's going on. Each episode is its own self-contained story.
Probably the only things you might need to know is that in the first episode, the main character, Meredith Grey (played by Ellen Pompeo), met and slept with Patrick Dempsey's character, not knowing that he would be her instructor at the hospital. They now are starting a relationship. Also, her mother, played by Kate Burton, has Alzheimer's and doesn't really remember her at all. Her mother used to work at the hospital where she's working, so a lot of the staff knew her mother, but none know that she has Alzheimer's. I don't know when she'll start telling people...or why she hasn't yet.
