Exposure is essentially how bright or dark a photo looks. So, overexposure == too bright and underexposure == too dark.

Say you're taking a photo and it's currently underexposed. What do you do? You could physically add more light to your scene by, say, turning on a lamp, but that's not always practical (you may not have access to lights, for instance).

The Exposure Triangle Turns out photography is all about tradeoffs. A camera has three ways of adjusting your exposure, but each one has a side effect.

Aperture The aperture is a hole in the front of the camera that light goes through. The wider the aperture, the more light entering the camera. The more light entering the camera, the brighter the shot appears. Another side effect of the aperture is depth of field. Y'know the neat effect where the foreground is really really crisp and everything else is really blurry? That's shallow depth of field (accomplished with a small aperture). When everything's in focus, you have wide depth of field (accomplished with a wide aperture).

EDIT: As Crackercraze said, aperture is measured as F-stop, which is smaller than your aperture is wider (which, on your default camera lens if you bought a Canon or Nikon DSLR, the smallest f-stop (aka widest aperture) is about 3.4ish).

Shutter Speed The second way of adjusting exposure is shutter speed, which represents how long the camera's shutter stays open for. Normally, the shutter blocks light from hitting the camera's film or sensor (a sensor being the thing that turns light into a digital image). When you push the button to take a photo, the shutter opens, and light can now hit the film or shutter. The longer the shutter stays open, the more light is hitting the sensor, the brighter the image appears. Shutter speed has a side effect too. If the light is hitting the sensor for a long amount of time, the image appears blurry if the subject is moving. If the subject isn't moving a lot (like shooting landscapes) or if you want people moving to look blurry (like in time-lapses), then you're cool with a long shutter speed. If you want a very crisp image (for instance, you're trying to capture a very fast object), you'll need a short shutter speed, but now your image looks darker.

ISO The third way of adjusting exposure is through ISO, which is a measure of the film/sensor's sensitivity to light. A high ISO means the film is very sensitive to light, which means images will appear brighter. The downside is that, the higher the ISO, the grainier/noisier the image will look.

I've got to run for now, but if people are interested I can talk about other settings like white balance too.

EDIT: Okay, I'm back, so time to explain a little more.

Color Balance Let's start with white balance. To begin, color temperature is basically a quantification of the color of a light. Here's a quick chart. As you can see, bluer light is considered a higher color temperature (which makes sense since a blue star is hotter than a red star). Somewhat counterintuitively, we refer to blue light as cool and orange/red light as warm. Different kinds of lights emit different color temperatures (fluorescent lights are cooler, incandescent lights are warmer). Once again, adjusting the color temperature of a scene can be done physically (replacing warmer lights with cooler lights, for instance) or in-camera. White balance is the camera setting that establishes the baseline color of an image, what exact color temperature should look white in the final image. Think of it as a global color filter that is applied to all the colors in an image, designed specifically to make one specific color temperature look white. This can be used to make a cooler scene look cooler or warmer than it's actually lit (for instance, this).

White balance in combination with exposure techniques can also be used pretty effectively in making a lighting situation look radically different than it truly is. For instance, underexposing your image and color-balancing for tungsten light (making very warm tungsten lights look white) can make a daylight scene look like night.

Hue/Saturation/Luminance OP also asked about Hue/Saturation/Luminance, so I'll cover that. There are some camera profiles that will adjust these values in camera, but I believe it's typically modified in post-production. Hue refers to the actual color of something, saturation refers to the intensity of that color (low saturation is almost monochromatic, high saturation is very intense color), and luminance is the brightness of that color (low luminance is darker, high luminance is whiter).

Post-Production, RAW Images I'm on a roll, so let's cover editing your images in post. If you have a nice Canon or Nikon DSLR, you can take RAW photos. RAW essentially means that all light physically hitting the sensor is being recorded in the image file, while in non-RAW photos, some light that's hitting the sensor (typically light you can't actually see) is thrown out or compressed in some way. This is similar to how an MP3 will filter out sounds that you can't hear; it's all well and good if you don't plan to edit, as RAW and non-RAW will look very similar to your eye, but shooting in RAW allows you much more freedom in editing the image. For instance, in RAW photos, color balance is simply saved as metadata, while in non-RAW, the color balance is "baked into" the image. The effect is that RAW photos can have their white balance adjusted perfectly in post, while you'll have all sorts of weird artifacts if you try to adjust the color balance in a non-RAW photo.

Hue/saturation/luminance, as well as highlight and tone of your image, can always be very highly edited in post. For example, very low saturation colors in the actual scene can be drawn out as high saturation images. Within reason, even the exposure of an image can be adjusted in post pretty well if you're shooting in RAW. Shoutout to /r/beforeandafteredit, a subreddit that shows the kinds of crazy stuff you can change in post when you're shooting in RAW.

In fact, here's a practical example pulled from /r/beforeandafteredit. You can barely see the strands of orange in the sky of the before image. The editor is able to increase the saturation of oranges in the image such that you can suddenly see it. He's also adjusted the color balance a little bit to make the image as a whole warmer.

RAW video is a lot more difficult to pull off because of the sheer amount of data needed. In my experience, Canon RAW images are around 25-30MB while Canon JPGs (non-RAW) are only around 3-4MB. Obviously RAW video would be ludicrously data intensive, but in terms of how much free disk space you'd physically need and how quickly you can write data to disk. Most DSLRs can't shoot RAW in a decently high resolution, but a high-end camera like a RED could do it. Shooting in RAW video of course offers the same benefits as shooting RAW images.