The idea is simple. Find samples, like the mummified passenger pigeon discovered recently in a museum desk drawer, and collect its DNA. Compare said DNA to that of its closest living relatives to see what specific genes make a passenger pigeon unique. Then splice those crucial genes into the living relative's DNA strands to produce a genetic copy of the extinct animal. Resurrection.
The restoration potential is not limited to plants and animals that we have just recently eliminated. We could also potentially bring back species like woolly mammoths or saber-tooth cats. Not dinosaurs though, since DNA has a half-life of just 521 years or so.
Of course, successfully bringing back the mammoth might also require restoration of its habitat, so it has a home to roam. But even without the reappearance of charismatic megafauna, such techniques will find uses from agriculture to injecting a bit more genetic diversity into dwindling populations of endangered species. The biggest contribution of the new biotechnology may not be de-extinction, but preventing extinction in the first place.
[The above text is a transcript of this podcast.]