A person is essentially a best friend. Although in Grey's Anatomy there are many best friends, only Meredith and Cristina have actually called each other their persons.
The person first came about when after Meredith and Cristina had a fight, Cristina revealed to Meredith that for her scheduled abortion, she had needed an emergency contact person. She put down Meredith Grey as her person.
When Cristina moved to Switzerland to run the Klausman Institute for Medical Research, Alex took over being Meredith's person, saying that Cristina has left him her shares in the hospital and, "She left me you, too."