Computer
Digital video is going to take A LOT of computing horsepower. While you can make movies on a mediocre computer, it will take you a LONG time to edit and render the video - so long that it might be considered cruel and inhuman punishment. If you already have a powerful computer, that's great, but for the rest of you... here is what I suggest.
CPU
If you can afford it, buy the meanest and fastest CPUs. These would be the Pentium 4 3.0 GHz and above, or the Athlon 64s 3000+ and above. If you just won the lottery then the P4 Extreme Edition or the Athlon FX CPUs are great choices.
RAM
Get at least one Gigabyte of the fastest memory your motherboard can use. You can never have too much memory when playing around with digital video. You can squeeze by with 512 MB, but you won't be able to run as many programs at the same time.
Hard Drives
Buy the fastest and biggest drives you can afford. Fast hard drives with at least a 7200-RPM spin rate are essential for smooth video capture and playback. Digital video will also eat up a lot of space. Hard drive prices are dirt cheap these days, so there should be no excuse to not get 200 GB+ sized hard drives.
Get two hard drives for your PC. Editing your video files will be MUCH faster when you are reading from one drive, while writing to the other drive. Serious video production with single hard drives is NOT recommended, as the drive has to continuously switch from read to write modes.
DVD Burner
Not really a necessity, but eventually you will want to give the videos to your friends and family. CDs, with their 700 MB of space, may not be able to hold your large video files. DVDs will be able to hold 4.7 GB, or about seven times more than a CD. In addition, you can archive your raw video files to DVD, thereby freeing up space on your hard drive.
External Drive
Just like the DVD burner, the external drive is also not a necessity, but it's great to have for backups. Hard drives do die, and it would be a shame to lose hours of work. Regular backups to another drive can avoid this.
Laptop
Should you buy a laptop? If you plan on editing on the go, then the answer is yes. I personally always travel with my laptop, because I have to edit and post my videos on the road. The single hard drive of a laptop will make editing/rendering much slower than on a comparable desktop.
Buy a laptop that can do better than 1024 x 768 resolution. Video editing tools need the extra space. You can also position multiple programs around the screen easier.
