Coffee contains caffeine, and caffeine has a number of documented side effects. If you drink too much coffee, you may experience symptoms of caffeine overdose. If you fluctuate in how much coffee you drink, you may experience caffeine withdrawal symptoms. For more information, see Caffeine & Health.
Beyond that, coffee can cause several types of side effects. One commonly noted side effect is that, for many people, coffee makes you poop.
For those with heart disease, drinking coffee may be less safe. Some evidence shows that drinking more than five cups of coffee a day is unsafe for people with heart disease. Additionally, drinking unfiltered coffee can increase overall colesterol, LDL colesterol and triglycerides, and occasional coffee drinking is suspected to have triggered heart attacks in certain people (though regular coffee drinking does not--it's the shock to the system caused by coffee that seems to be the issue).
Apparently, some people use coffee for enemas, which is unsafe. I don't even know what to say about that except, "Why would you choose to do that??? Stop it!"
Drinking less than two cups of coffee a day appears to be safe for pregnant and breastfeeding women. Drinking more than that appears to be risky.
Children are more susceptible to side effects from coffee than adults. It is not recommended that children be given coffee with full caffeine.
Caffeine may also exacerbate anxiety disorders, bleeding disorders, diabetes (due to additional fluctuations in blood sugar), diarrhea, irritable bowel syndrome, glaucoma, high blood pressure and osteoporosis (due to an increase in the flushing out of calcium).
Still, all that said, coffee is generally considered to be pretty safe when consumed in moderation!