What happens between making a DNA library and screening by nucleic acid hybridization? I understand how a plasmid library is made and how each clone is put on the nitrocellulose membrane and nucleic acid hybridization is done. What I don't understand is how each clone is "identified" and separately put on the membrane?

The library is 'created' during a ligation reaction: plasmid vector + insert (e.g. cDNA). That DNA is used to transform _E. coli_ , and, after plating out the transformation mixture, single bacterial colonies are obtained. Each colony contains, to a first approximation, a single plasmid species (i.e with a single insert). So each colony represents a clone, in the terms that you use.

I don't know if anyone really does things like this any more, but originally each plate would be transferred to nitrocellulose, for probing, and to a fresh plate for archiving. This way, any colony identified as positive after nucleic acid hybridisation could be picked from the archive plate for further characterisation.

This screening can be done on large format Petri dishes to reduce the number of plates needed. I know of one lab which in the 1980s used to screen λ libraries after plating out for plaques on cafeteria trays. That was the heroic era of molecular biology.