Aspirin, like many drugs, is a wonderful drug and a horrible drug. It is used prophalytically, meaning to prevent, for a MI (heart attack) and for strokes, and many other reasons such as muscular pain/cramp. It isn't without it's side effects and SHOULDN'T BE TAKEN IF UNDER 18 YEARS OF AGE.
The biggest and most concerning side effect of aspirin is it reduces your clotting factors. Now, what does that mean? It means that say you have a paper cut, when you are taking aspirin it's going to take longer to stop the bleeding and for your blood to clot. What's the big deal? This can be an issue with certain people and in certain circumstances. The most obvious one is surgery, you want your blood to clot at a therapeutic rate when you are being 'cut open'. If someone takes aspirin often or in large amounts that increases there risk of hemorrhaging (bleeding out). Obviously tests are done to prevent this (INR, protime) but that's a risk. Decreased clotting factors also means more susceptible to bruising, which can either just be an annoying side effect or serious. If a person got into a car accident and was on aspirin, there is a good chance they aren't going to just have internal bruising, it's internal bleeding.
A more complete list of side effects can be found on a medical site such as mayoclinic.com or webmd.com. I suggest to take side effects with a grain of salt, depending on the medication and the length of time you are on it, the side effects can be truly minimal. With the being said, you are taking a medication that is changing chemical components in your body and you should always be informed as to how it works, why your on it, the side effects, and any other pertinent information.