Allopatric speciation is when you have one population which becomes geographically isolated into two distinct populations. Over time and through specific mutations and adaptations to their own environments, these different populations become genetically distinct enough to become their distinct species.
Adaptative radiation is when you introduce a species to a new area for colonization. Over time, different populations of organisms will begin to adapt to specific parts of the environment and over time and through many mutations helping them for their specific part of the environment, they diverge into a distinct species.
Darwin's finches is an example of adaptive radiation. There were three different types of seeds. Large, medium, and small. Different beaks evolved amongst a population of finches to exploit each distinct type of seed. Over time, these populations became distinct species-- all as a result of the different seeds.
This is a rough overview of both and I can go into more detail if you need me to.