TL;DR: For now, we simply don't know. To date, we have never seen more than a few.
At the moment - as of A Dance with Dragons (books) and Season 5 of the show - we know very little about the Others/White Walkers, their society, or even their origins. Most of what we do know is in-universe speculation & rumor from various characters. In either source, however, we have never seen more than a handful of the creatures at once.
Origins (Reproduction vs. Transformation)
Neither the books nor the show has given a definitive origin for the creatures, called "The Others" in the books and "White Walkers" in the show. However, episode S03E04 (Oathkeeper) of the show has definitively shown that
Craster's sons are transformed into Walkers via the Night King's touch.
However, we do not know if this the sole method of reproduction or simply another way of adding to their ranks. What we do know from the books is that The Others have existed on Westeros since before the arrival of Men, and they may even pre-date the Children of the Forest, which would make them the oldest species in existence.
We also know from the books that The Others possess genders just like Men. Although we never see a female of the species, the story of The Night King describes one:
A woman was his downfall; a woman glimpsed from atop the Wall, with
skin as white as the moon and eyes like blue stars. Fearing nothing,
he chased her and caught her and loved her, though her skin was cold
as ice, and when he gave his seed to her he gave his soul as well. He
brought her back to the Nightfort and proclaimed her a queen and
himself her king, and with strange sorceries he bound his Sworn
Brothers to his will. For thirteen years they had ruled, the Night’s
King and his corpse queen, till finally the Stark of Winterfell and
Joramun of the wildlings had joined to free the Watch from bondage.
After his fall, when it was found he had been sacrificing to the
Others, all records of the Night’s King had been destroyed, his very
(A Storm of Swords, Chapter 56)
However, this story ALSO states that the king was "sacrificing to The Others", which could indicate that he was delivering babies like Craster. But given that females exist and that the king "gave her his seed", we also have to assume that they can reproduce sexually. The most likely scenario is that females of the species are extremely rare - that would then explain why the transformation method is used to augment the species.
Also, since we only saw the first few seconds of the process, we don't know if the transformation method
transforms the infant into a baby Walker which must then grow up, or if it somehow turns him into a fully functioning, adult Walker.
To date, we've been given no indication as to the number of the species. They have armies, yes, but the ones we've seen have been hundreds of Wights for every one Other/Walker. At most - during the scene in Oathkeeper - we've seen fourteen Walkers together.
Since the bulk of the species is said to live in the deepest parts of the North where no Man has been, it's impossible to speculate how many Others/Walkers there are in total. For all we know, there could be entire cities of them in the coldest areas.