Speciation takes place when one species diverges and becomes two separate unique species. The key to this is that by general definition, two different species cannot produce fertile offspring.
Speciation occurs usually over time through natural selection as a result of genetic mutations and conditions which favor particular traits.
Adaptive radiation is one way in which speciation occurs. It begins with one common ancestor which diverges through speciation to become several different species which are adapted to different environments. In adaptive radiation, the species remain closely related.
One of the most famous examples of the idea of adaptive radiation are the finches which Darwin discovered in the Galapagos Islands.
Because each island has different conditions the birds developed different beaks which helped them to find food in particular conditions on particular islands.
