Step 1: Download and install a torrent client. Takes about two minutes total. µTorrent is actually so light it doesn't even need to be installed.
Step 2: Download a torrent file of the anime you'd like to download. These are tiny, generally under 50kb.
Step 3: Double click said torrent file and your client will start downloading the thing. New releases can usually be downloaded at the highest speed allowed by your connection, so if you're on DLS or Cable it usually goes really fast as most releases are 170-200Mb.
Also, many fansubbers nowadays like to make two version of their releases: XviD and h264 encodes. H264 is more processor intensive so old computers start lagging when viewing them, but the image quality tends to be far superior to the older XviD, so that's what you should aim for if you have a decent rig.
Also, if you don't have the correct video codecks on your computer, you won't see anything when trying to run a video file. In order to get everything you need in a neat package, get the
Combined Community Codeck Pack. With CCCP you also get Media Player Classic, which most people agree is
the best media player for anime. It is very user friendly, doesn't require installation and eats minimal system requirements, which is good when watching super high quality encoded videos that your system might not be able to handle on heavier players like Windows Media Player.