Asteroids are rocks of varying sizes that can be found anywhere in space. Their main purpose is to provide you with different ores and other useful items (such as lava, glass, glowstone, nether wart, etc.). Remember to bring a spacesuit while mining as there is still the danger of suffocating!
Asteroids made of stone are the most common type you will find. They may contain any ores found in vanilla worlds, usually in above vanilla quantities, as well as ice and different variants of stone(granite, diorite, andesite). It is important to note that each asteroid contains at most three types of ores, so if you can't find what you are looking for, try moving to a different asteroid.
Nether asteroids are very much like little slices of the nether. They contain netherrack, soulsand, quartz, nether bricks and red netherrack. Nether warts can also be found here, but you shouldn't worry about any of the mobs usually found in the nether: they can't spawn on asteroids.
Ice asteroids are almost entirely made out of ice and packed ice. The only other things you can find on these barren rocks are pieces of obsidian, sea lanterns and, occasionally, some diamonds.
Glowstone asteroids have an exterior made out of stained glass and glowstone, while their cores are filled with lava which spills over the edge. The only resource that can be gathered here is the glowstone which is found in above vanilla quantities.