I haven't posted much on these boards in a LONG time, but I was directed here by Johans Nidorino to help out a little bit. (I'm "CrappyCaptureDevice" on YouTube and I make videos showing where elements in Super Smash Bros. Melee and Brawl come from. So I do record video from my Wii, from Brawl.)
Anyway... I'm more or less going to duplicate some of the information that has already been said, but sometimes it's good to get different perspectives on the same thing.
As Fryguy64 and 8bitretroshit were saying, you'll need some type of video capture device. You will either install it in your computer in a PCI slot or something or if it's outside the computer, it is usually hooked up via a USB port. Then, you will hook some type of video input (e.g., satellite receiver, VCR, Wii, whatever) into it, and you will interface with it via software to record video into a video file on a hard drive in your computer. There are a lot of different capture devices, and as Fryguy64 said, some aren't gonna be that great, other will be better. (You can say that about anything, really.) [If you download television shows not long after they are aired, this is how they are recorded into a video file so you can download them.]
With that being said, there are some other things you'll need to consider. As 8bitretroshit said, you're gonna need a fairly decent computer and a hard drive with a lot of free space. Having a dedicated slave drive for capturing is a good idea.
Another thing is how you're gonna play the game as you're recording, since most capture devices have you hook the video source directly into them, and most have a really small preview window in the software and/or lag, so... as 8bitretroshit was saying, you will most likely need to have some type of splitter, so you can play the game on your TV while you are recording. So, I hope your computer is in the general area of your television.
The next thing you might want to start looking at capture devices to get an idea of what you might want to get. Again, like anything else, there are a lot of choices. I got my capture devices from NewEgg.
www.newegg.com/Store/SubCategory.aspx?SubCategory=47&name=TV-Tuners-Video-DevicesThis category has some things in there that aren't capture devices, like HDTV antennas the size of flash drives. So you'll want to look at only the devices which are designed for capturing video. I would say this is a good place to start looking.
You said you wanted something that's of "relatively high quality." Considering that the Wii only does standard definition, you should be fine. So really, I think that as long as you're not too much of a tight-wad (spend AT LEAST over $50, if not over $75 or $100), whatever card you get should do a pretty decent job, as long as you got everything set up correctly.
Now, if you're wanting to capture 480p (60 distinct frames per second), that's a bit of a different story. You might have to get something similar to what I have. (Explained below.)
It's kind of difficult to, from scratch, explain everything that's involved in capturing video game footage. So, start doing some research (Google, etc.) and start asking some questions and I will do my best to help you out. (As I'm sure 8bitretroshit and others will too.)
As for me, the capture device I use is the
Hauppauge HD PVR 1212.
Yeah, I did pay over $200 for it, but it does a nice job, can record in high definition, and I don't have to worry about splitting my video signal, since it has video loop-through with no lag, which is really nice.
8bitretroshit:
I watched the Brawl match you uploaded to Megaupload. I gotta say that the sound quality is really good, but, have you been deinterlacing your videos? (Since I know you said this video was really old.) Here's a screenshot from your video of what I'm talking about:
You might already know this, but the lines in areas of movement are from the
interlacing. Though the Wikipedia article is a bit technical, what interlacing basically is... is transmitting two fields per frame, with each field being a different point in time in the video. One field will take up the odd numbered rows of pixels and the second field will take up the even numbered rows of pixels. It's basically a cheap way to get two points of movement in one frame. (I'm really glad we're moving away from this to progressive scan.)
To get rid of this, you'll need to
deinterlace the video. There a lot of different ways to do this, some better than others.
This site explains it some more, though I can tell you a faster and easier way to do it than everything it suggests at the bottom, if you want.
If you have a capture device that can capture progressive scan, then you don't have to deinterlace, since every frame is complete and doesn't have interlacing in it. Basically, progressive scan is having 60 solid frames show up a second instead of 30 interlaced frames.
My capture device can record in progressive scan. For a little comparative analysis, I captured a match on the same stage, Delfino Plaza, and also uploaded it to Sendspace. (Since Megaupload was erroring out on me.)
Sendspace Link. I encoded the video in x264 format with mp3 sound. So, you should have the x264 codec installed to play it. (I would think most of you would already.) Or use VLC Media Player. (I encoded it so the filesize would be about the same as yours, even though mine is only a little over a minute long...)