Beer mats are flat, multicoloured pieces of cardboard. Usually, they have an advertisement for a beer on them. In England they are often painstakingly peeled apart by frustrated drinkers or used for a game of 'Flip' during boring moments.
Instructions for 'Flip'
Balance a mat on a table, and make sure that half of it hangs over the edge of the surface.
Hit it with with the back of your fingers from below (this can hurt your nails if you strike it particularly hard).
Try to catch the beer mat without turning your hand over.
Beer mats are very rarely used for soaking up beer drips as was the original intention.
On the European continent, however, they have a different purpose entirely: keeping track of how drunk you are. Each time you order a drink a notch is marked onto your beer mat and then at the end of the evening, the bar staff will add up your notches and you can settle your account all in one go. This is a very trusting system that nobody seems to abuse.
All of this is illegal of course, as the European Parliament ruled that no beer mat should be used twice, something about it being unhygienic. So if, when you take a seat at a table in a pub in Europe, you find a beer mat there that's had one side peeled off, has got some doodlings on it and a complete stranger's phone number, or indeed any other signs of previous use, then you will know that that the pub you are in is breaking the law. You will probably find such beer mats in most pubs in Britain, as the English don't seem to like the European Union very much.
Beer mats are also collectors items. Not necessarily in the same way as antique china vases, but more in the way that a train's engine number is valuable to trainspotters. People have been known to show off collections of beer mats that number in the thousands. Not such a bad hobby if you are into drinking beer as well, but most beer mat collectors don't touch anything stronger than milk.
It's not only advertisements for beer that you will find on beer mats though. Brandy, vodka and cigar logos have all featured on beer mats. So too have sporting events such as the football world cup, the rugby world cup and various horse races.