The primary risk of aspirin, as with other NSAIDs (nonsteroidal anti-inflammatory drugs) is the risk of internal bleeding, in particular from bleeding ulcers. NSAIDs work by inhibiting hormones known as prostaglandins. Inflammatory prostaglandins are released when the body is injured to dilate blood vessels in the area to increase oxygen and nutrients to aid in healing. Excess dilation of the blood vessels though leads to leakage from the blood vessels, which cause the swelling and pain. So NSAIDs like aspirin work by constricting blood vessels to prevent blood vessel leakage. This is also part of the reason NSAIDs can increase the risk of heart attack and stroke, as well as lead to kidney failure, liver failure, tinnitus, loss of vision, etc.
Peptic ulcers from the use of NSAIDs result from the inhibition of another prostaglandin needed for the production of the stomach's protective lining. Without this protective lining stomach acid attacks the stomach wall leading to ulceration. When bleeding starts the anti-clotting effects of the aspirin increases the risk of death from internal bleeding.
It is estimated that NSAIDs kill an average of 16,500 people a year, with aspirin accounting for about 5,000 of these. This does not include the non-fatal injuries such as chemical induced hepatitis and kidney failure. Examples of other NSAIDs include ibuprofen (Advil, Motrin, Nuprin), celecoxib (Celebrex), and naprosyn (Aleve).