When you're under the weather, you feel sick. It can also be a good excuse: "I'm sorry I can't visit your grandmother with you, but I'm a bit under the weather today."
There is one popular theory about the origin of under the weather that connects it with sailing. Supposedly, when a sailor wasn't feeling well, he would retreat below deck, out of the "weather" above, where he could be described as under the weather. Other experts aren't sure about this explanation and suspect the phrase has more to do with the old idea that the weather has a strong effect on people's health.