The Holy Ghost is God.
I don't think anything in Scripture suggests anything about exactly how it happened except that it didn't involve a man inseminating her. My personal thought is that it involved minimal interference with human biology, i.e. something very close to what's known to be possible scientifically.
Certain traumatic processes are able to induce cell division in lagomorph ova. Clearly a haploid mammal could not be gestated to term, so Jesus cannot have been diploid - human non-diploid cells arecancerous or hydatid moles. Now, the euarchontoglires are a clade of placental mammals including both lagomorphs and primates, so the closest relative of ours which shows some sign of being capable of parthenogenesis diverged from our ancestors some time in the Cretaceous, specifically between 85 and 70 million years ago. That isn't that distant a relative, so i'd say primate parthenogenesis is not completely impossible.
The very simplest explanation is, rather startlingly, that Jesus wasn't conceived at all but was female! I don't think this is impossible either. However, if he was male, it could be concluded that he was genetically identical to Mary except for the Y chromosome, which was of the Cohen Modal Haplotype because he was born of David's line. It's just about possible that he was "conceived" via a viral infection.
But, i think what happened was that a process akin to the trauma which lagomorph ova undergo to begin mitosis occurred to one of Mary's secondary oocytes. This can be achieved by physical damage or sudden cooling. Some kind of physical process like that is the most conservative explanation.