Ahem. Well, see, that's the trick. It's just like recoding music, you see. You want the best picture you can possibly have before you record so you have to do as little as possible with the resulting frames. A good camera is like a good mic and a good lighting arrangement is like a good recording room. Without those, you're starting handicapped and it's hard to compensate with even top of the line tools. There's no Beetmaster equivalent for video.
Let me ask you a question. Are you familiar with Photoshop or Photopaint? If so, then take a frame from your footage into the program. If you can improve the quality to your satisfaction of that one frame within either of those programs, then it's possible to improve the movie either through a good video tool set or through a script for either program. If not, then probably no tool will help you.
There are filters that can remove grain, if that's your problem, or..
There are filters that can balance color, pretty common problem, actually and a lot of ways to adjust it.
You can change day to night pretty easily in a variety of ways
You can remove jitter (or add it) with a motion tracking tool.
You can soften or smooth an image with blurring tools.
If your image is pixelated, a tool that removes grain might or a smart blur might help.
There are no tools that you can use to add information that wasn't there originally. If the picture is blurry, you cannot make it sharp. You can make it more blurry. If the picture is overexposed, you cannot undo that. You can make it less bad, perhaps, by changing the way the black and white clip. Keep this in mind when to make your video.
Make sure you are lighting enough. Lighting is a big deal when shooting video. It makes all the difference in the world. A good lighting rig is nice to have if you plan to make a lot of videos, but you *can* mix a few of the big florescent lights of different colors (yes, color matters) from Home Depot and get yourself well lit but not overexposed and make do with that. It's a ton easier to work with later in the editing stage and. Just don't over expose the subject.
Remember, few digital cameras are going to give you 24p - that "clear" film look that high end videos have. Those videos are shot on film, that's why they look so good. It's the 24p plus fabulous lighting. Low end digital and even most prosumer models don't come close to getting this look.
There are things you can do with consumer camera video (besides selling it on craigslist). One thing you can do is work the camera's style into the video as if it was intentional. You can also mix the live video with higher quality photos, frames, simple cartoons or motion graphics and blend the two, or put the footage into a vintage TV.
You know that people aren't going to be wowed by your camera, so make the story interesting to watch. That's why they're watching anyway.
Yep, read what Trezza said. No software can compensate for bad lighting and a cheap camera. Lighting is the most important aspect of getting a quality image. With all the consumer/prosumer mini-DV cameras out right now, you can achieve amazing results with proper lighting. On the flipside, you can have a 3 CCD HD camera with poor lighting and come up with absolute crap. Don't waste time on programs that "improve quality". Prevent the problem from the beginning before you go into post.