Becoming infected by a website is possible, but most web-based malware involves third-party code, not a passive exploit. Unless you're running an ancient browser, you're going to be prompted before it runs anything like that. First of all, you won't need to run third-party code to download a torrent. So don't do it. You can minimize the chances of web-based malware by downloading the torrent directly from a reputable torrent indexing site, like Pirate Bay. Make sure you don't click on the plethora of fake "download" links and ads. Finally, if you're running a modern OS like Vista or Windows 7, don't run your browser as administrator or under an administrator account.

My money would be on executable files in the torrent. You shouldn't have to run an EXE to watch a movie. A legit release will just be the media file, like AVI, and a readme. If you're pirating software, watch out for any other executables like keygens or installers, but in that case you're going to have to at least trust the cracked software. Look for releases from a "reputable" group at the original source, like a torrent or newsgroup posting.

Academic point: media files *can* cause viruses or other malware if they exploit the software that is reading them.

Yes, the vast majority of infections come from directly running an exe file or a tainted installer. But everyone knows not to run weird exe files. That's where exploits like buffer overflows come in. They can use malformed data to trick the program into running arbitrary code.

That said, that type of thing is extremely rare and usually patched quickly. Keep your media player up to date and you should be fine.