This is a new drug that is given as a pill twice a day to treat cancer. This is one of the new, "smart" drugs. It binds to a protein on the surface of the cancer cell called VEGFR, and this way it slows down the growth of cancer cells and kills them. Head and neck cancer cells are known to carry this protein on their surface. Research in animals and in patients with other kinds of cancer showed that Axitinib can be effective at killing cancer cells, or stopping their growth, by this mechanism. It is generally a safe drug that is given by mouth. The investigators do not know, however, whether Axitinib is effective in head and neck cancer. This research study is being conducted to learn if Axitinib works in head and neck cancer, and also to learn to predict who would benefit from it. Four blood draws will be done to check special blood tests while the subjects are treated with Axitinib. These will be drawn at the same time as your routine labs, and there will not be additional sticks needed. A biopsy of the tumor before and after 1 month of treatment may be obtained to test how the cancer cells are responding to treatment. By testing these blood and tissue samples, the researchers will look at special tests (protein molecules) to try to determine what kind of head and neck patients would best respond to this drug. This is an open-label study, meaning that all subjects are on the active drug and there is no placebo (sugar pill).