During the days when ships were powered by sail, the captains log documented everything that happended during the day. As sickness could spread rapidly on a ship,there were often times where the number of sailers that were ill exceeded the space provided in the log to record their names. During these times, the excess names of the sick were recorded in the next column, which was reserved for the weather conditions of the day. Thus, it was not unusal for an ill sailor to be listed "under the weather".
Ryan: "Jesus, who's that chick you were with the other night? I sure would like to pound that pussy."
Nikko: "Yeah, she's hot, for sure, but she's a little...how shall we say.... "under the weather upstairs".
It means to feel sick. In most cases, it's used to say that you feel a little sick. ... But… you usually feel low on energy and feel tired when this happens. To be Sick and ill both mean to feel unwell.
This is the British middle class equivalent of the exclamations 'ill' or 'sick', meaning cool. Being terribly modest and reserved, one doesn't want to overstate things.