Polymer clay is a vinyl-based craft modeling compound designed to be cured in a home oven. The clay will never harden on its own when exposed to air, but will cure into a hard, slightly flexible and dry form when baked at low temperatures --- usually around 275 degrees Fahrenheit, depending on the brand. You can also bake and cure polymer clay using a toaster oven or kiln oven with low enough heat settings.
Oil-based modeling clays, most commonly known by the brand name "Plasticine," are designed to stay soft permanently and be used in developing sculpting skills rather than creating permanent projects. If you try to place oil-based clay projects in an oven, or apply heat to them in any way, they will simply melt. Therefore, do not use this clay for any permanent projects.
Earthenware air-dry clays are water-based, and thus designed to become hard through exposure to air. However, you can use an oven to help them harden, not because heat is required, but because heat encourages faster evaporation of the water. To take advantage of this, set your oven to a heat below 200 degrees Fahrenheit and place the clay item inside. Leave the door slightly ajar to allow the venting of moisture as it evaporates from the clay. Monitor the clay closely; check it every 10 minutes to see how dry it is. It will be hot to the touch, but you can tell how it's drying by watching for it to change color and become chalky in texture.
Paper clay is another form of air-dry clay made largely from finely ground paper fibers. As with earthenware air-dry clay, you can use an oven to help dry out this clay. With paper clay, however, it's even more important to monitor the clay against scorching. Use the lowest oven setting possible, and check your clay items every five minutes. Position the clay on racks as far from the heating elements in the oven as possible.
Pottery clay is also an earthenware clay that you can use a conventional oven to help dry, but you must use a pottery kiln to cure the clay. This is because the process of transforming clay into ceramic requires intense heat, strong enough to melt and fuse the silicate within the clay; this is much like the process of making glass. A conventional oven simply doesn't get hot enough.