Theoretically, mammoths could be cloned by recovering, reconstructing or synthesizing viable mammoth DNA and injecting it into the egg cell of a modern elephant whose nuclear DNA has been removed; alternatively, mammoth genetic material could be introduced into an elephant genome in order to create a mammoth-elephant hybrid or chimera.
This raises an ethical question as to whether we should start the journey down one of these paths.
Habitat disruption
Some people worry about the match between an extinct clone or novel chimera and the modern natural habitat. Firstly, there are worries that introducing these new species could disrupt natural ecosystems, or the so-called "balance of nature".
This was one of the many ethical themes in the novel and film Jurassic Park. Perhaps extinct animals are dead for a good ecological reason, so to speak, and "unnaturally" re-introducing them, particularly those that vanished long ago, would be futile or even pose serious global risks to life on the planet.
This is a common concern that should be addressed. Mammoths were widely around until 10,000 years ago, with some surviving until as recently as 4,000 years ago. Mammoths are thus not particularly 'alien' organisms from a modern ecological standpoint.
They lived and evolved alongside much of the modern flora and fauna, including humans - who are, incidentally, likely responsible for their extinction. It is therefore highly unlikely that the introduction of a mammoth population would severely perturb natural ecosystems.
In any case, the extent to which there really is a "benevolent balance of nature" has been vigorously contested in biology. Food webs are often maintained despite significant fluctuations in extinction, invasion, migration, diversity, and energy pathways.