Infamy means being famous for something bad or negative. You may be hoping for fame when you get an enormous tattoo of your favorite pop star on your back, but there's a chance you'll end up with infamy instead.
The noun infamy is most often used to talk about famously evil or terrible people or historical events. The day the Japanese attacked Pearl Harbor, just before the start of World War II, was described by President Roosevelt as "a day that will live in infamy." Infamy contains the root word fame, but rather than meaning "the opposite of famous," its meaning is something closer to "fame gone bad."