iOS: There are a lot of ways to monitor your PC's BitTorrent downloads from your phone, but if you want to actually download files to your device, iTransmission is a real live iOS BitTorrent client that will do it for you.
While torrenting huge files seems a little silly on an iPhone or iPad, it makes sense for certain things, like music — especially if you don't have a computer around to download, import and sync with iTunes. For some reason, the developer has pulled it from Cydia, however, so you'll have to download the .ipa file from its Google Code page (linked below) and install it manually. To do so, you'll need to install Installous from the cydia.hackulo.us repository in Cydia. Then, download the iTransmission IPA file from the link below, and double-click on it to open it up with iTunes. It should sync to your device just like a normal app bought from the App Store.
iTransmission works like most other torrent clients. Just find a torrent on the net, copy the link to the .torrent file or copy its magnet link, and hit the plus sign in iTransmission to add it. It will download the files to your phone, which you can then retrieve later with something like iFile, which you can get from Cydia. And, just like desktop BitTorrent clients, you can set speed limits, automatically forward ports, and edit how many people can connect to you. You can also download from 3G or Wi-Fi only, if you don't want to go over your data cap (by default, it's set as Wi-Fi only).
iTransmission is a free download, works on iOS 4 and above. Note that it's still in beta, so a few features aren't yet implemented, but overall it works quite well for downloading torrents to your phone.