A monopod, as it's name suggests has only 1 area of support, and it can never be as stable as a 3-legged tripod. However, there are some monopods that have retractable "feet" at the bottom that splays out so that it's more stable. But they're still not going to be as stable as a tripod.
Like some have said, they're great for use in tight places, and where you need some support with good control. Off hand, I can think of concerts, open-air shows, sporting events (especially with telephoto lenses) etc - basically any place with lots of people. A tripod occupies a lot of space when deployed, and it's too easy for people to knock against it and tip your expensive kit into the ground!
Monopods are light and easy to pack, and make excellent companions to a IS (image stablised) or VR (vibration reduction) lenses.
Monopods are easy to lug around and you'll probably get to use it much often than a tripod if you like shooting everything around you. I love mine so far, I can hand-hold a 1/20 shot at night with a 50mm lens with reasonable sharpness that I can't get without a monopod (even if I rest my arms/elbows on something firm).
Needless to say, shots that requires a second or more will require a tripod (and probably a remote release), but I find tripods cumbersome to set up and use while walking around town.
Monopods, beside being lighter, smaller and more mobile, allows for quicker setup (as compared to a tripod) so it is useful if you will be constantly on the move and do not want to miss that important shot.