Malts are the basis of all-grain brewing. They're made when the kernels of a grain (usually barley) are allowed to sprout and grow for a few days. They're then heated to stop the growing process and convert some of the starch inside into sugar. You can read about the process more in depth here.
The way the grains are heated makes a huge difference in the qualities of the malt. For the most part, barley is basically the same (although there's a hotly contested difference between two-row barley and six-row barley).
The real way malts stand out from each other is how long, how high, and in what way they're heated after germination.
Kilned malts are a very popular style of malt, and they're heated in a very different way. Rather than heating them with water to convert the sugars inside, they are heated with the intention of drying them out. Some are dried out at a lower temperature (sometimes as low as 100F) and some at a higher temperature (as high as 220F). This gets rid of most the of the water leftover from the germination process, usually bringing the grains down to 3% to 5% moisture.
Why the wide range of temperatures? It's all about a balance of diastatic power and flavor. Barley kernels contain diastatic enzymes, which are used in the conversion of starches into sugar.
It's what you need for mashing to work, which is why some specialty malts can't be mashed - the diastatic enzymes are destroyed by the high heat. Basically the lower the temperature at which the malt is kilned, the more diastatic power it will have.
This is why with many kilned malts the drying is conducted at a low enough temperature that at least some if not all of the diastatic enzymes survive.
This means that the malt can convert its own starch to sugar during the mashing process and must, in fact, be mashed for the conversion to take place.
With that high heat that kills the enzymes, however, comes an impressive range of flavors that emerge naturally in toasted barley. These flavors can be toasty or bready or simply that classic "malty" taste. The higher the temperature at which you kiln a grain, the more pronounced the flavors will become
You don't want to turn down those flavors just for the sake of diastatic enzymes, but you do need them. That's why beer recipes call for different types of malt. Those low kilned malts that still have all their diastatic power are often called base malts. They have very little flavor of their own (though they do have some), but they possess the ability to convert all their starch to sugar during the mashing process. Because of this, a grain bill (a recipe's list of the different malts it requires) is usually mostly made up of base malts, with much smaller amounts of other varieties.
This is because the other malts just don't have the diastatic power to convert their own starches, and they need the base malts to help them along.
A certain amount of variety also comes from air circulation, or the amount of air that is allowed to move around the grains as they dry. Sometimes the grain is given a lot of ventilation as it's kilned, drying it out more quickly, while sometimes it's given less ventilation and allowed to dry out much more slowly.
Between heat and ventilation, there is a very big range among the varieties of kilned malts. Here are a few of the popular styles.
Pale malt - Pale malt is a very basic kilned malt. It is heated at the lowest temperature of all the malts, usually between 100F and 120F for as long as 24 hours. This low and slow kilning process dries the barley kernels out without sacrificing any of their diastatic enzymes. Because of this, pale malt (also often called pale ale malt) is an extremely popular base malt and is called for in a slew of recipes. Of all the base malts, it imparts a little bit more color, usually scoring between 3 and 5.5 on the Lovibund scale, which measures the color of malt.
Vienna malt - Vienna malt is another that's kilned at a relatively low temperature, though it can be heated as high as 160F. Despite the heat, it usually retains enough of its diastatic power to convert its own starches during mashing. Even so, it's usually called for in small amounts along with big bulk amounts of base malts (with a few 100% Vienna exceptions). It is known for its toast or biscuit like flavor and the pleasant orange color (around 10 Lovibund) that it imparts on the beer.
Munich malt - Munich malt is kilned at a higher temperature (between 195F and 220F). It also has enough diastatic power to convert itself, but it doesn't have any to spare and can't be used as a base malt. It has a sweet, bready flavor and imparts a nice amber color at somewhere between 10 and 20 Lovibund.
Aromatic malt - Aromatic malt is kilned at a similar temperature to Munich malt. It is especially sweet and gives the beer a malty, almost syrupy flavor and aroma. It has some diastatic power and is usually able to convert itself, but can't be used as a base malt. Even apart from its enzymes, it usually makes up less than 10% of a grain bill because of its very strong flavor and color.