Color spaces are the containers into which we put images to define what they look like. Files do not simply exist with a color, they exist with a set of numbers and how those numbers are translated into color depends on the display device and the related color space. One example of how this is usually described is a movie seen on multiple TV screens in a store. The color and contrast of the image depends on the setup of the TV set, and "messing with the controls" can change the appearance of the image of a particular set even as all of the sets receive the same picture.
Most cameras offer the option to capture in sRGB or Adobe RGB. sRGB is the most common color space, and the one we noted earlier as the space in which our monitor lives. Output devices like the color printers at labs are also sRGB, so they are in the same key. But, even as two trumpets need to be "tuned" to sound alike, the printer and the monitor function in the same color space, but still need to be tuned, or "profiled" to be compatible with each other, and make beautiful music, or output prints that match each other.
sRGB - A relatively small color space (gamut, or range of reproducible color), but recommended for most people who do not have a specific reason to work in a larger space. Casual shooters and portrait and wedding photographers working with outside labs typically work in sRGB. Working in a larger color space like Adobe RGB or ProPhoto and sending files to be printed in sRGB will result in dark, flat and desaturated colors as the expanded color range will be clipped off rather than used for printing. If you use a lab for your printing you should know what color space they prefer you to work in. The sRGB color space is usually safe.
AdobeRGB (1998) - a larger color space containing a wider gamut of color, and the space preferred by offset printers, and therefore used by most digital image editors and photographers whose output will be reproduced in print. Many cameras allow you to capture your images in the Adobe color space (jpgs) instead of sRGB. Most photo quality inkjet printers have a sufficient gamut or range of reproducible color to take advantage of the Adobe color space. It is therefore a good choice for those making their own prints with better desktop printers. Note that raw files do not have a specific color space until acted upon by the converter, where they are considered to be in the largest color space, ProPhoto.
ProPhoto - The largest color space generally accepted to include all light visible to the eye. Raw files opened in Lightroom and the ACR Adobe raw converter are initially in a very wide gamut color space similar to proPhoto until the output file converts them as determined by the settings you choose. It is also natively a 16 bit color space capable of holding all the available information from a raw capture. This allows greater editing freedom without banding and other artifacts caused by clipping color.
Color management is how you deal with the way an image travels from the source, through the editing, to output. Color management can be relatively hands off unless your output goes in more than one direction. Most people working with a digital camera may send images to a printer like Costco, and email images to their friends and family. Perhaps an image will end up on the web in a personal web gallery. All of these outputs can be dealt with within the sRGB color space. How well profiled your monitor is will determine how closely your images will match those of someone else with a profiled monitor. Without profiles all bets are off.
A test of your system is pretty easy. Send a file to a printer, like Costco, and they will print it like a normal lab, chasing the color and density of the image as the machine sees it. Ask them to print the file with "auto off" and you will see the print as you prepared it with your software. If it doesn't match your monitor (within reason) it is your fault, not theirs. Send the same file to a friend in an email or post it on the web. Now, view the file on someone else's monitor, and look to see how it compares to your original file and the prints you had made. This is a visual representation of color management.
Differences are inevitable, but they should be relatively minor, and not something that makes you wonder what happened to your great photo. The print is your better comparator, as it should appear pretty close to what you created on your monitor. The further apart the appearance of the image on your monitor and the print, the greater the evidence that you need more control over color management.
Your goal is to have the neutral values, overall density, color and contrast appear pretty much the same everywhere. If things are really out of whack, you need to look for the flaw in your settings to see where you can make changes and get into line with everyone else. If you work in more than one color space, like Adobe for printing on a good desktop printer and sRGB for email and outside printing, you need to pay closer attention to making sure your files are in the proper space when they go out of the computer. Start in the larger space (Adobe) and make modified files in the smaller space (sRGB) as needed.
Color settings in your software package are also important. Setting your color space and other preferences in Photoshop, or telling Elements you want to make your own decisions is the first of many steps in controlling your images. Turning color management off is simply not an option (well, it is in the Elements dialog box, but not one you should choose).