First and foremost: the spider plant (ribbon plant, airplane plant, St. Bernard's Lily, malamadre/"Bad Mother," spider ivy, hens and chickens) is incredibly easy to propagate, and can be propagated a variety of ways, including the harvest and sprouting of seeds from seed pods.
1. The most common method of spider plant propagation is water propagation, both because it is easy and because it is entertaining to see/watch root growth through clear glass containers.
When propagating in water, you first want to take a pup from the stem of a mother plant. You want a pup that is relatively hardy, and preferably beginning its rootball growth. With this pup, you will want a water receptacle that will not drop the pup directly in the water. I like glass bottles, such as the ones from soda or coffee, but the possibilities for the glass receptacle are nearly limitless. (Tip: darker glass inhibits algae growth!)
The only part of the pup that should be submerged is the butt/beginning of the rootball. Leaves left in the water will rot. After as little as two days, you may begin to see root growth on your young friend. After a few weeks, the pup may be ready to plant in soil. A good rule of thumb is to wait until the plant has at least 1 - 2in. of root. I prefer to wait until the roots begin to branch off, as well, but this is possibly up to personal preference.
I'm told that you can also use a moist sponge or damp perlite compound to sprout pups. They love the water, so long as their leaves are not constantly submerged.
2. Another means of propagating the spider plant is by taking a mature spider plant pup with a defined rootball, removing it from the mother plant, sticking it in soil, and watering it thoroughly. This is probably best for a spider plant pup with a rootball about as thick as your thumb, or more.
Keep the soil that your pup is planted in moist, and don't be afraid to spritz your pup with water, once in awhile. I've found that spider plants seem to particularly enjoy a "shower" of this sort.
Planting directly in the dirt is much like a game of "survival of the fittest," in my mind. The hardier your pup, the more likely it is to survive this kind of direct planting.
3. An interesting variation of the direct planting method is to house a spider plant pup in it's new container (or the dirt next to its mother plant), then pin the stem down instead of removing the pup from the stem. This would allow the pup to situate itself while still under the care of the mother plant, and likely increases the survival rate of directly planted spider plant pups.
Again: keep the dirt moist. Spider plants (particularly C. comosum) are native to tropical and South Africa, and are happy to give you good growth under moist conditions. This, too, is probably why they seem to like spritz baths as much as they do.
Once the pup has acclimated itself, and developed roots, clip its umbilical support stem, dividing it from the mother plant, and move to its own little spot. It may be a good idea to keep an eye on the newly snipped pup for awhile, however well it seemed to have integrated itself into its new spot, though.
4. Another method of spider plant propagation is division of an overgrown parent spider plant. This method is best down out-of-doors.
Simply take the spider plant (or plants) out of its container, knock as much dirt off the roots as you can (without doing too much damage to the root systems), and then (with sharp knife or scissors, generally speaking -- I've done this the sloppy, bare hand way, myself) divide the plant where sub-sections of the plant itself reach the roots, and keep the roots with the sectioned off plants, as much as you can.
Don't freak out too much over some root damage. In fact, cull any roots that look as though they might be damaged, and then dispose of both purposefully culled root bits and accidentally culled ones.
Replant your divided spider plants in individual containers, or put a few of the sub-divisions in new containers while putting the initial plants back in their old home (or homes).
Best of luck with your spider plant propagations!