What is compression?

Or more specifically, what is image compression in a camera?

Basically compression describes the way in which your image is stored on the memory card (CF card) in a more compact way that takes up less memory space, because the original uncompressed photo would take up a LOT of memory. You can have two main compression styles of storing your photos:

(1.) Lossless Compression with RAW photos
Lossless compression means that  you don't lose any information in the storage of the photo. The image that you save will contain exactly the same information as the original uncompressed image that was snapped, only it takes up about a quarter of the memory space as your uncompressed image. You can set your camera to take lossless images by setting it to take RAW photos, which are usually in RBG TIFF or just TIFF format. The problem with RAW images though is that, understandably, they take up more memory space on your memory card than lossy compressed pictures. For example, a RAW image may be around 6-4 MB in size whereas a lossy compressed picture can be 3.0 - 0.2 MB in size.

(2.) Lossy Compression with usual photos
Lossy compression, as the name suggests, is accompanied by some information loss, so that the picture you end up with is not exactly the same quality as the original uncompressed image. The difference is seen when you blow up the picture and inspect it under high magnification, although for most functional purposes of photos, you probably wouldn't usually need to ever magnify the picture to this extent so the information loss won't really be noticeable under normal circumstances. The format of lossy compressed photos is usually JPEG.

Note that once you convert something to a JPEG file and lossy compression takes place, it is irreversible and you cannot restore the raw quality from this file.

With lossy compression, you can control the image quality to be 100% quality and all the way down to 20% quality if you want (although the latter probably isn't advisable!).

How do you control the compression on your camera? 

Most cameras allow you to control the compression.

On my camera you can control how fine the lossy compression is (i.e. whether the JPEG is 100% quality or lower) by entering the camera Function menu and selecting compression. My camera gives you options for superfine compression, which gives the best quality shots (marked by S on my camera screen), high-to-normal (marked by a quarter circle image on my camera screen), or normal quality shots (marked by a step-like pattern on my camera screen). I tend to set mine to superfine for normal shots or if I know I'm going to do an artistic shot, I may want to set the image to RAW lossless compression and compress it further later (with a program like Lightroom or Raw Therapy) if needs be. I guess professional photographers would work more with RAW images. On my camera if you want to take a RAW shot you shoot the image and before it saves you can disrupt the save by twiddling the zoom lens as it saves, click the function button and set it to RAW.