I agree with Greg S here, "The day after tomorrow" has no real science behind it. The fact that your teacher put this in a science lesson says more about your teachers understanding of science than anything else.
Basically, the movie is about climate change, but only a little. The writer(s) read a little about climate change and thought that they should make it more impressive to sell as a disaster movie. A lot of scientists who work in this area are quite angry that the movie was made because they have a serious message to give people and the movie makes it seem a little silly (I seem to remember ice actually chasing someone, which is fine in a movie about magic but not in a movie which is meant to have a scientific basis).
As a basic introduction to the science put some ice blocks in a glass and then fill it with water, this represents ice floating in the sea. While the ice melts the level of water stays the same because the ice was already in the water (this is Archimedes's principle, google it if you haven't heard of it before). If you partially filled the glass with water and held the ice above it in a spoon or something instead then the ice represents ice on mountain tops and on arctic islands. In this case the water level will rise as the ice melts because the ice was not in the water to start with. When this happens in the real world you get floods in places like London because it is near the sea and the river Thames. Scientists have good reasons to think that things humans are doing will make the Earth warm up, slowly but steadily, so these floods could happen.
By the way if you do that experiment, do it in a sink because there is a lot of water involved.
That is only an introduction, and it is a very, very complex bit of science. So nobody, can really be certain quite what will happen, I don't think there are any scientists who think that something like "The day after tomorrow" will happen. On the other hand there are also very few scientists who don't think that climate change is very important and dangerous.
If you want more information about climate change try to get your teacher to show "An inconvenient truth" or google it.
Finally, remember there are a lot of people who make money out of doing the things that scientists say causes climate change and these people do not like the idea of climate change being important or caused by them. Also there are several nut cases out there. Since almost anyone can publish stuff on the internet they can all say one thing or another about climate change. So, if they don't have a lot of different sources that all make sense, you don't have to trust them.