How does this work?

sorry, this is so elementary, but i have no techy apititude at all. when i
just click on the name or the mirror of the show i want my computer says it
doesn't recognize the file type. (.torrent)
is there something i'm supposed to download first before i can download these
shows?
and once i download them, do they stay on my computer, or is it a streaming
video that turns off when i close the window?

sorry for the elementary questions, but i've never used something like this
before.
thanks!!

the wikipedia bit torrent page is probably a good place to start. i'd bet that
utorrent probably has a good faq as well.