De-extinction is the restoration into the world of living specimens of animals that had once been extinct. There are three technologies for de-extinction currently being explored, each one with a target species its project hopes to de-extinct.
A. Back-breeding. This is only possible where there exist living species with genetic homologues to the target extinct species. By this method, the desired genes are to be found in another species, and a selective breeding program, combined with gene reading (to confirm which genes are present) is studiously and strictly employed. There is a project under-way that aims to de-extinct aurochs. Aurochs are the ancestors of domestic cattle, and had inhabited Eurasia and North Africa. The last known auroch died in 1627. This technology requires not only for there to be living relatives of the extinct species, but also for them to largely contain the genetic material of the extinct species.
B. Cloning. This technology utilizes somatic cell nuclear transfer (SCNT) to transfer the genetic material from the extinct species into an egg cell of a related living species. The modified cell is then moved into a host which allows the embryo to develop into a living creature. There is currently a project that aims to de-extinct the Pyrenean Ibex, which has been extinct since 2000, and which is closely related to the other three sub-species of Spanish Ibex. Although the principles of SCNT are established, it is still a hit-and-miss venture, likely due to the enormous number of tiny factors necessary for proper development. Thus, out of hundreds of attempts to clone the Pyrenean Ibex, only a single foetus has been brought to term, and which died minutes after birth due to lung abnormalities.
C. Genetic engineering. This is done by editing the genetic code of a living species which is relatively similar to the extinct species. There is a program underway to de-extinct the passenger pigeon which became extinct in 1914 due to massive hunting. Until that point it had been one of the more abundant bird species in the world, and was considered to be the cheap meat.
Caveats: It should be noted that neither back-breeding nor genetic-engineering would produce a species that has an exactly identical genetic code to the extinct species. While cloning would produce a genetic twin to a member of the extinct species, it's not clear exactly what that would achieve. One animal does not a species make. And for all three methods, the produced animals would not have the same epigenetic make-up, microbiome, or even "culture" as their extinct predecessors.
There are objections to all of these projects: (1) Animal welfare, (2) health risks (e.g. as novel viral hosts), (3) environmental risks (e.g. if released to novel environments with unexpected consequences), (4) political impact (e.g. change perception of what "extinct" means, or view de-extinction as frivolous use of public funding), or (5) moral arguments.
Similarly, there are benefits expected from these projects: (1) knowledge (insight into extinct species), (2) technological advances resulting from the work, (3) environmental benefits (e.g. rescue damaged ecosystems that had depended on the role of the extinct species), (4) justice (i.e. to make up for the sin of their extinction. ie. where humans are to blame), and (5) wonder (i.e. the awesomeness-factor).
What to do? Each person needs to ask this of themselves. The question can be broken down: Should de-extinction be publicly funded? Should it be categorically banned? Should it be regulated?