This episode of SciShow highlights three methods to "resurrect" extinct species...
Cloning. Animals can be cloned using a technique called "nuclear transfer". Simply put, genetic material from an extinct animal is transferred to the egg of a surrogate animal. (Then, cross your fingers and hope for the best.)
Genetic Reconstruction. Let's say you only have fragments of DNA from the animal you want to clone. No problem. Take those fragments and piece them back together with "filler" DNA from a closely related species. (Jurassic Park, anyone?)
Backbreeding. This technique essentially "breeds" an extinct animal back into existence. Take cattle, for example. If you know the specific traits of an extinct breed, you can "resurrect" those traits by selectively breeding existing cattle.