Sharing mate (pronounced mat-ay) is a strong tradition in Argentina. Yerba mate is an herb which is typically served in a gourd with hot water poured gently over top. A bombilla is essentially a metal straw with a filtered end which is used to drink the mate. The taste is a tad bitter and if you're new to it your host or matero may sweeten it with a bit of sugar.
The matero holds the thermos of hot water and traditionally takes the first, and strongest, drink. Each person takes turns drinking, passing the mate back to the matero each time so that it can be refilled and returned to the next drinker. Once you've had enough simply say gracias, which in this case means a polite 'no more thanks'!
Mate has a mild stimulant effect, not unlike caffeine but with a less pronounced high and low. In Argentina people often tote a thermos and mate with them wherever they go and hot water vending machines are common.