Fig.: to be sick

I'm feeling a bit under the weather. I'm probably coming down with the flu

Non mi sento benissimo; probabilmente sto covando l'influenza

I'm sorry I have to call our dinner off. I have this constant and annoying cold and I feel under the weather

Mi dispiace ma devo annullare la nostra cena. Ho questo continuo e fastidiosissimo raffreddore e non sto proprio bene

The origin of this idiom is not clear, but it seems realistic to assume that it's a term of nautical origin.
According to some, on the high seas, during times of rough seas and bad weather, sick passengers and sailors would go below deck, which would hopefully protect them from adverse conditions. So, they were literally "under the weather".
According to others, in the days of sailing ships, captains logged all crew members who were ill in a column. If there was no room left in that column, they were written in the "weather" column, which gave "to be under the weather" the meaning of being sick