Adaptive radiation is the relatively fast evolution of many species from a single common ancestor. Adaptive radiation generally occurs when an organism enters a new area and different traits affect its survival. An example of adaptive radiation is the development of mammals after the extinction of dinosaurs.
In evolution adaptive radiation is the idea that you have this relatively rapid or fast expansion, the development of many new species coming from one original common ancestor. They called radiation because the species if you're drawing a diagram radiates form an common ancestor. One very common way for this to happen is when an animal moves into a new area that has lots of new ecological opportunities. the classic example of this is what's known as Darwin's Finches. When Darwin explored the Galapagos Islands he found a variety of different birds very obviously related to each other but each uniquely suited to the different island it was on. In some islands, there were say beetles as the major source of food. So the beaks of these finches were designed to best eat beetles. Whereas others you may find your food in the form of nuts and so they had to have a different shape beak designed to crack the nuts to get at the food inside. And so as the bird came over and expanded throughout the Galapagos Islands, the chain of islands, created by the volcanic action in that area, they all originally derived from some mainland bird but few of the got blown over the one island say for example those who had the best traits to survive in that island lived, those who had the traits that didn't allow them to survive died and therefore didn't pass on their traits. And ultimately the population there got well designed for that first island. Then maybe they flew off to some other island, a storm blew them, something happened that allowed a few of them to migrate to yet another new island and had to adapt to the selective pressures that were on that island. And this just allowed you to create lots of different kinds of birds from one starting population. Now again I say relatively first. This is the geological time scale. You may be talking about only 500, 000 years or 30, 000 years. It's not fast as in it happens in now, alright?
Another way this commonly happens is when you have a mass extension event, wipe out a large number of organisms thus leaving open whatever it was those now extinct animals were doing and allows other animals to take their place perhaps exploit new resources that hadn't been exploited before. Or ultimately when a new evolutionary trait is evolve, that allows new opportunities that just didn't exist before. Like when flowering plants first developed they had greatly speeded up their ability to engage in sexual reproduction because they took advantage of the terrestrial terrestrial animals and used them to spread their pollen. Now a very common example used to represent this is what happened to the mammals after the dinosaurs died out. Mammals had been around for well over 100 million years but prior to the extinction of dinosaurs they were relatively small minor group. But after the dinosaurs died out, all of a sudden there was lots of new opportunities. If you were a little fuzzy ball and scrap like thing, you could hide in the bushes and the dinosaurs wouldn't see you. But once the dinosaurs were dead, the bigger ones who would have attracted too much attention from ones who would've been dead, now the bigger ones that we're mutants they could expand into their new ranges into their new ecological niches and start developing lots of different ways and that's where we get the diversity of mammals that we see today. Whether it's humans with our wonderful hands and brains or bats with their wings or even wolverines with their claws and skeletons.