No, it does not copy or store actual fingerprints. The fingerprint scanner systems compare specific features of the fingerprint, generally known as minutiae. Typically, human and computer investigators concentrate on points where ridge lines end or where one ridge splits into two (bifurcations). (See the graphic in this section)
The scanner system software uses highly complex algorithms to recognize and analyze these minutiae. The basic idea is to measure the relative positions of minutiae, in the same sort of way you might recognize a part of the sky by the relative positions of stars. A simple way to think of it is to consider the shapes that various minutia form when you draw straight lines between them. If two prints have three ridge endings and two bifurcations, forming the same shape with the same dimensions, there's a high likelihood they're from the same print.
To get a match, the scanner system doesn't have to find the entire pattern of minutiae both in the sample and in the print on record; it simply has to find a sufficient number of minutiae patterns that the two prints have in common. The exact number varies according to the scanner programming.
(a) Gray-scale Fingerprint (b) Minutiae points.