Alex & Izzie: OFF
They were going on dates but not officially a couple when Izzie met Denny Duquette, one of her patients. Since then she and Alex have not been seeing each other at all.

Bailey & her husband: ON
Although he almost died at the end of season 2 in a car accident as Bailey was giving birth to their baby, they are both alive and together.

Burke & Cristina: OFF
Burke left Cristina at the alter.

Chief & Adele: OFF
The Chief & Adele split up but it is obvious that they still care about each other deeply.

Chief & Ellis: OFF
The Chief and Ellis Grey had an affair many years ago when Meredith was a child. Ellis wanted him to leave his wife for her, but he did not. The Chief visited her sometimes as she had Alzheimer's, until she recently passed away.

Denny & Izzie: OFF
Denny proposed to Izzie. Then he died later the same day. They loved each other, and Izzie is so distraught by his death that she quit that hospital the day he died. He left her over 8 million dollars, but she hasn't cashed the check yet.

Derek & Addison: OFF
Derek & Addison have finally gotten a divorce. Addison cheated on Derek with his best friend, and he caught them together, which drove Derek to move to Chicago in the first place.

Derek & Meredith: OFF
They are currently not together, but Derek recently told Meredith that he loves her. When Meredith was choosing between him and Finn, he said he was walking away, and when she told him she broke up with Finn, he hardly reacted. Derek and Meredith seem to have an on-again off-again relationship, but haven't been on for a while at this point. They're regularly either a couple, broken up and mad at each other, or just "friends," but probably secretly in love with each other. When Derek moved to Chicago, he had a one night stand with Meredith the night before their first day of work. She often refers to him as "McDreamy." Just recently, Derek broke up with her because he thought she was committed to the relationship.

Finn & Meredith: OFF
Meredith broke up with Finn for Derek. She has referred to Finn as "McVet."

George & Callie: ON
George and Callie got married.

George & Meredith: OFF
They slept together one night after George had had a crush on her since their internships started. Meredith started crying during, and George got mad at her for doing that. Meredith decided she had made a huge mistake. After a long period of not speaking to each other, they are now friends again.

The Interns

Meredith Grey
Cristina Yang
Izzie Stevens
George O'Malley
Alex Karev

The Doctors/Nurses

Derek Shepherd - In love with Meredith Grey. Divorced Addison.
Preston Burke - Previous love interest to Cristina Yang. Recently shot and then had surgery to regain full use of his hand.
Miranda Bailey - Sometimes referred to as "The Nazi." Married. Has a baby. Very direct personality.
Addison Shepherd - Derek's ex-wife. Cheated on him with his best friend, Mark, causing him to move to Seattle.
The Chief (Richard Webber) - Had an affair years ago with Ellis Grey. Separated from his wife, Adele. Chief at Seattle Grace Hospital, hoping to retire (the obvious choices are Derek Shepherd and Preston Burke).
Callie Torres - George's wife. It happened really fast.
Olivia Harper - Nurse. Had a short fling with George until he found out that he got syphilis from her because she caught it from Alex. Caused problems with Izzie and Alex's relationship at one point when Izzie caught them in bed together. So she and Alex seem to have some sort of at least sexual relationship at points.

Other

Ellis Grey - Meredith's Mom. Had an affair with the Chief when Meredith was young, causing her then-husband Thatcher to leave her. She had Alzheimer's, but recently passed away.
Joe - The bartender who owns the bar across the street from Seattle Grace Hospital.
Denny Duquette - Patient who Izzie fell in love with. He had a heart condition which he ultimately died from. He left Izzie millions of dollars in check which she has yet to cash in.
Finn Dandridge - The vet who had obvious romantic intentions with Meredith, but she chose Derek instead. They met when Meredith and Derek brought him their dog because he was sick. He is often known as McVet.
Adele Webber - The Chief's wife. Threatening to leave him if he doesn't retire.
Thatcher Grey - Meredith's dad. Left her and her mom when she was young because of Ellis' affair with the Chief.