St. Martin lived in the fourth century AD in Hungary. He was born in 316 or 317 and was later to become the bishop of Tours, but in his youth, following his father's wishes, he became a soldier of the Roman Empire. St. Martin's Day is celebrated on the 11th of November, which is the day he was buried after he died in 397 (coincidently, this is also the day Fasnacht starts - see here).
Now as the story is told, one day in winter, Martin (or Martinus by his Roman name) came across a poor, half naked man, freezing to death in the cold. Now Martinus was a soldier, sitting high up on his horse - it would have been easy to overlook the poor guy, but he didn't. He wanted to help, but he didn't have anything on him but his weapons and his military uniform. So he took his sword, and cut his warm, bright red coat (which historically was probably white, but red sounds nice in the story) in half, offering one half to the guy on the ground.
Now in his honour, there are parades everywhere where children will march through the dark holding lanterns (often hand crafted - we used to make them all together in kindergarden, later my mum helped me, and the last one I personally made a few years ago can be seen here) and singing special songs about St Martin, his deeds, and lanterns. They also act out the scene with the beggar or, if they're lucky, there's an actual horseman to do that. At the end of the parade, children will get pretzels or special rolls (Martinsweck). I don't know if they do that everywhere, but when I was in kindergarden and elementary school, we would only get one pretzel for two kids, so we had to share it, just like Martin shared his coat. Here's a video I found on Youtube that gives quite a nice impression of a St Martin's parade:
Another custom that is quite common, but we personally never did this, is to have goose for dinner on St Martin's day. There are several stories behind this, but the one I heard as a child goes as follows:
When Martin had left the army, he started living as a kind of hermit, but he soon gained quite some followers for his good deeds. Eventually, the people of Tours wanted to make him their bishop, but, being humble as he was, Martin refused. The people insisted, and he fled, hiding in a geese-coop. But the geese, being, well, geese, quacked and chattered loudly, people found him, and he became the bishop anyways. So today, we eat them - you know, to thank them. Or maybe to punish them for giving poor Martin away.
Now there are many more traditons surounding this holiday (which isn't actually a day off by the way, you only get to celebrate in the evening) - for example, in some regions the kids go from door to door with their lanterns, like on Halloween - but these are the ones I have personally experienced and grew up with. It was one of my favourites as a kid, because crafting the lanterns was always so much fun, there's a horse, and the songs are really beautiful - and today, I still like the lanterns, still like horses and still love the songs, but over everthing else I love this thought of sharing and caring. Some people are now trying dispose of Martin's Day because, being a Christian celebration and all, it's deemed inapropriate to celebrate in public school where children of other faith's are exposed to it, but I really hope they keep the tradition alive in one way or another :)