Most cameras will let you pick them up and start shooting right away. Set the camera to Auto, indicated by a Square, the word 'Auto' or some other picture (all dependant on camera type, but almost always green) and it will adjust the settings for you. A lot of the time 'right' as well. Which is to say they generally get the shadows dark, but not pitch black, and the whites bright, but not blown out entirely. A good number of things are in focus, and - if it's light enough where you are - not blurry.
Certain times you're in a specific situation. Night Portrait, (Day) Portrait, Beach, Snow, Sports/Action, Landscape... and your camera probably has a settings for that too. Indicated by a semi-descriptive picture and the words on the screen when you go through them. Some really go all out and have a rather lengthy list of custom scenarios. (On a beach, at night, while it's snowing and you're lit by candles being slightly too specific, but they're getting close)
But enough about those. They don't really need an article. P Tv Av M. Or possibly M A S P do.
Picture via Flickr - You'll notice I leave out A-DEP, but so do a lot of cameras
Let's go through them from 'easiest' to 'hardest'.
P - Program
When you'd like a little more control over your camera than is possible with the auto function, start here. It unlocks a lot of things like ISO (sensitivity to light, just like film, higher is more), over- or under- exposing the shot, different file options to save (RAW, if the camera has that feature) and so on. Unless you change anything, however, it does tend to act like Auto - in that it does do a lot of the calculations for you. The difference being, you now have the room to nudge them in a certain direction.
Tv or S - Time Value or Shutter Priority
Next on the list, as it does what it sounds like. This is handy for action shots, or wildlife (children included ;) ) that won't stay still. Set the value to 1/200 second (or more) and you can freeze a lot of things that would otherwise look like a blur. The camera still has the room to adjust the aperture - or how much light comes into the camera. The tradeoff being that the faster you set the camera the less number (or more accurately, less depth) of things end up being in focus. Force it to go extremely fast and you'll notice a very dark picture. Ooops. Learning from mistakes is just as valuable as learning from success.
Av or A - Aperture Value
This is where things get a little odd. All you really need to remember is that the smaller this number is, the less of your image will be in focus. Conversely, setting this number higher will result in a lot more things (or specifically depth) being sharp. As always there's a downside to pushing the extremes. The higher you set this value, the longer the shutter has to stay open. Which can lead to movement blurring or past a certain threshold, less sharpness overall. It is, however, quite handy for extremely closeup shots - as the closer something is to the camera the less 'depth' is actually in focus. More on that in an upcoming article.
M - Manual
While the previous two modes had the camera adjust for the 'missing' value, here everything is up to you. You'll notice that if your camera has an option for over- or under- exposing the shot, it, almost paradoxically, vanishes. This is because everything is under your control - so it will be bright, dark or just right based on your settings -- with no help from the camera. As you might guess, this does make things a little tricky. I would very strongly recommend a few test shots in the other modes first so you'll at least have a starting point. This mode is most often used (at least by myself) when the other two modes overcompensate, or vary the 'missing' value and I want to override that.
Example #1: Taking the first shot in some other mode and forcing the same settings for the following shots - with the intent to stitch them together afterwards. Otherwise you could find a wide variance in exposure or amount of the image that is in focus from image to image.
Example #2: 'Forcing' a larger region of focus AND having a short shutter time - at the expense of image brightness - often used when something is both close to you and moving. This is when you can find the other modes are falling short. If the speed is fast then not enough is in focus OR it leaves the shutter open too long (and a blurry image) when adjusting for the focus alone. While I could also go into Tv/Av and say I want an underexposed image, I wouldn't know by how much. It would be easier to set both the Av to what I want (say 5) and the shutter speed to something useful (1/200s).
Apart from spinning the dial, it does vary quite a bit from camera to camera what buttons change the values in question. Do check your manual. If you've lost it, most camera manufacturers also make a copy available online.