There are two main forms of protection used on thermal protection systems for reentry vehicles today: ablative and thermal soak. Ablative is the most robust, and it's what was used on all the Gemini and Apollo vehicles*. Soyuz uses such a system, as do Mars landers such as the recent Mars Phoenix Lander. The reentry capsules from Genesis and Stardust, which returned to Earth recently, used ablative heat shields. Broadly speaking, the process of material burning off of the shield forms a protective layer around the vehicle, shielding it from the worst of the heat. The downside is that an ablative shield has to be very thick (since it's constantly losing material, and you don't want to run out before you get through peak heating) and can only be used once. For a vehicle like the Space Shuttle, which is a) meant to be reusable and b) very big, it's simply impractical.
The alternative is to come up with some kind of material which is lightweight and able to form that protective thermal barrier all by itself. Silica-based ceramics are great for that, soaking up huge amounts of heat without conducting any of it to vulnerable structures behind. A single tile of the Shuttle's TPS is lighter than a roll of toilet paper, and can be easily crushed in your hand. And it soaks up a LOT of heat; one party trick NASA likes to do for the media every now and again is to stick a tile in a furnace until it glows red-hot and then pull it out with bare hands. It does have its limits; it's great for a long, slow reentry but would fail at high peak loads. (By "fail" I mean that too much heat would penetrate through it, melting the structures behind it.) On the Space Shuttle, this is handled in two ways. First, the areas that experience the most heating get reinforced carbon-carbon (RCC) panels, which work differently from the tiles but are much heavier. Second, the reentry profile is designed to spread the heating out more generally so that no one area gets too hot. It is important to emphasize this; the Shuttle's TPS is very good for what it does, but constrains the Orbiter to a very tight operating envelope. A small error in the reentry profile and it will overheat.
There is a third way to protect from heat, and that's radiative cooling. The Shuttle's RCC panels use this, as did the earliest Mercury capsules (the suborbital ones), which had beryllium alloy heat shields. ICBMs also rely on this method to protect their warheads during reentry. The materials for such heat shields are generally very expensive and also relatively massive, which is why those methods aren't generally used for large systems. RCC has another problem: it's brittle. The Columbia accident unfortunately demonstrated that its even more vulnerable than anyone realized, and of course any area with a hole is completely useless. It also needs time to cool and radiate the heat back; the Shuttle's banking maneuvers give it regular opportunities to do so.
Check out the Wikipedia article on reentry (
Atmospheric Reentry) and also the
Space Shuttle News Reference Manual. The former is a general article; the latter has a lot of technical stats and details on how they are made.