a file is hosted by a person on their computer, they are called a peer.
The peer creates something called a torrent file, information file about what and where the file is located etc...
That torrent file is then uploaded to whats called a tracker. There are tons of tracker sites, many fake bullshit, out there.
People use a client software, (program that will download, upload, manage and decode the torrent and its files)
Again tons of programs, Utorrent is the one you would want to stick with, if you get into the more private torrent tracker sites, they require something like utorrent, whole other subject to talk about why.
So you download this torrent from the tracker, utorrent will run and begin downloading pieces of the files hosted on the peer. At the same time your computer will begin uploading those pieces to other people who are trying to download the same file from the peer.
When a download is complete on someone else's computer, they too become a peer and do whats called seeding.
The more seeds you have on a torrent, the faster it will be disbursed to everyone trying to download it.
That is a very basic laymans understanding of a torrent file.
As for the last question on how to play the files you download, it depends on what you downloaded, music, movies, apps, games, rar's pars, zips, 7zip exe etc...