If you're asymptomatic, you don't show any signs of being sick. In some cases, you can have a disease but still be asymptomatic.
When doctors talk about someone being asymptomatic, they are usually referring to a patient who has been exposed to an illness or is sick but doesn't have any symptoms. This is important in medicine because these asymptomatic people can often spread the illness, even though they don't appear to be sick. The a- prefix here means "not," from the Latin ab, "away from."