Photos coming out too light or too dark?

The "problem" is probably your camera's exposure meter.

Cameras are simple devices and one of the assumptions they make is that the overall tone (how light or dark the overall photo is) will average out to a set medium tone (~18% gray, which is about the same tone as the skin on the palm of your hand). If large portions of the photo will be darker than the assumed tone, the camera will think the reason for that darkness is lack of light and it'll brighten it up to make up for it. And the opposite is also true -with extremely light-colored scenes, the camera will think that the lightness is due to too much light, and it'll make them darker so they come out gray.

So, if you have a dark scene that you're shooting and you want to tell your camera that the scene should be darker than that "average tone", you have a few choices: #1. If you want accuracy, you can buy a "gray card" to calibrate your camera to the lighting conditions, set the exposure, and take the photo. #2. To get pretty close, you can meter off your hand. Or, #3, you can use trial and error like I do most of the time -just use exposure compensation in whatever mode you're shooting in:

Exposure compensation uses the exposure graph you see and use in manual mode or the "priority modes", it looks like:

[-2...-1...0..+1..+2]

..(Note that on Nikon's it's reversed and the positive numbers are on the left). There's usually a small marker under the numbers that indicates where your exposure is on the graph using whatever settings the camera is on. The "0" on the graph indicates what the camera thinks is proper exposure (assuming that "average tone"). Setting the exposure towards positive numbers means you're telling the camera to add light or brighten the photo as compared to what the camera would usually choose. And negative numbers, then, darken the photo as compared to what the camera would choose to be correct.

So, if the camera is making your snow scene turn out gray (it's coming out too dark), you need to brighten the photo, or expose more towards positive numbers. Try exposing at +1, then adjust up and down from there until you get the result you desire. On the other hand, if you're taking a photo of your black cat lying on a dark blanket, it'll likely come out too light, so you'll have to expose towards negative numbers to darken the photo. Try -1, then adjust up or down from there.

Experiment:
If you want to see just how wrong your camera's meter can be, find an all white subject (a wall or your ceiling should work), and an all black subject. Take a photo of each. Lighting shouldn't matter, but you can try adjusting the lighting to prove that it has no effect -just make sure the lighting is consistent over the entire area that you'll be taking a photo of. You can add subjects to your photo as long as they too are 100% white or black (you'd want to take a photo of a white subject against a white background, then a black subject against a black background). If you want to take it even further, you can switch subjects and backgrounds and take 2 more photos (white subject on black background and vice versa) -those should come out mostly correct since the two tones (one dark, one light) will average each other out. My photos are here if you wanna cheat ;-)

Your 100% white and 100% black photos should all come out almost the same shade of gray no matter if your subject was all white or all black. -That's what your exposure meter is designed to do! Most of the time it works just fine, but when you have a scene that's made up of extremely light or dark tones, remember this and use exposure compensation to make the photo come out as you see it instead of all gray.