Playing around with liquid nitrogen is a pretty dangerous thing and I wouldn't recommend actually coming into direct contact with it! (A safer alternative might be to ask your science teacher to make ice cream using liquid nitrogen to rapidly cool down cream and milk ... very tasty.) I did a little web searching and found a few accounts of people swallowing liquid nitrogen as a party trick (I'm not sure how true these accounts are though) and most were not good.

Swallowing nitrogen gas traps it in your body because your epiglottis (the flap at the top of your esophagus) prevents gas from escaping. The result can be burns throughout your esophagus down to your stomach. Gas also expands and so it can push on your internal organs and resulted in a collapse lung in one guy who tried drinking liquid nitrogen.

There is an effect called the Leidenfrost effect that some people quote as being protective against liquid nitrogen burns. The Leidenfrost effect refers to a really warm object coming into contact with something really cold, resulting in evaporation and creation of a gas cushion that insulates the warm object from the cold object. For example, if you get a frying pan really hot and then sprinkle water onto the surface, the water beads up and jumps around the pan as opposed to just evaporating on the spot. This is because as the water hits the hot pan, there is some vaporization of the liquid drop which creates a cushion of gas below the drop. This gas cushion can actually suspend the water droplet and cause it to skitter around the pan. Gas doesn't conduct heat as well and so it doesn't evaporate as quickly. So in the case with liquid nitrogen, it is possible that you could get that protective layer of nitrogen gas between you and the liquid nitrogen. However that protective layer is pretty thin. In the case of people swallowing liquid nitrogen, that layer was not thick enough to protect their gastrointestinal tract.

Drinking liquid nitrogen is a lot like "swallowing fire" or putting big flaming torches in your mouth. It is very dangerous unless you know exactly what you are doing. What happens when you drink liquid nitrogen has to do with how much you drink.It is technically possible, but very dangerous, to sip a tiny amount of liquid nitrogen. A few drops poured straight into the back of your throat in theory should immediately boil because of the rapid change in temperature and humidity when they leave the surrounding liquid N (around -320F) and enter your mouth (around +96F). If this works right, the nitrogen will evaporate before it hits your skin. If this doesn't work, for instance if you use enough to maintain a "cold-core" of liquid N by the time it hits your throat, the liquid N will freeze your skin solid, causing a burn which will immediately destroy all of the cells frozen.