Huderon has explained it fairly well, but there is one key difference between "normal" downloading and "torrent" downloading.
With normal downloading you get a file from ONE place, so, as huderon says, it is a sequential down load, get the first byte first and then go through, step by step, to the last byte.
If the connection to this ONE place is slow, or a lot of people are trying to download it from this ONE place, then YOUR download will be slow.
With torrents you download a file from LOTS of places.
So if you want to download a file, it may be on 100 other people's computers.
So the torrent program starts taking parts of it from one person, parts of it from another person, and in the end it may be taking parts of it from all hundred people.
And after taking parts of it from all these different people it re-assembles it on your PC as one complete file.
The great thing about torrents is that even if some of these 100 people have their computer turned off the torrent program can still download the file from the 50 or 75 who may have their computer still on.
The other great thing is that because the download is being shared by 100 people it will not be held up by a slow line or if a lot of people are trying to download it.
The other great things is that if YOU start downloading a file, and even if you only have small part of it, YOU can start sharing it with other people, so YOU become one of those 100 people it can be downloaded from (at least the parts you have).
This is the process used by filesharers who want to distribute a large file like a film or a complete music album.
While it can be used to download pirate movies etc torrent also has a genuine use to help speed up the distribution of large files like in say online TV broadcasts (like say the BBC iplayer)