To do blue-white screening, you use a vector that contains the genes for ampicillin resistance (ampR) and the enzyme ß-galactosidase (lacZ). If you put the vector into cells by itself the cells will be resistant to ampicillin and will produce ß-galactosidase (which splits lactose into glucose and galactose).
The cloning site, or insertion site, is in the lacZ gene. This site is the sequence of nucleotides recognized by the restriction enzyme used to digest the DNA that contains the gene you want to clone.
Remeber that when foreign DNA and the vector are cut with the same restriction enzyme (assuming it produces sticky ends) the restriction fragments can insert into the vector (creating a recombinant vector).
Since the insertion site is in the lacZ gene any restriction fragment that manages to recombine with the vector will disrupt the lacZ gene and the plasmid can no longer code for ß-galactosidase. Like a really big insertion mutation.
So you mix the restriction fragments you got by digesting the foreign DNA with your restriction enzyme with the plasmid vector that you also digested with your restriction enzyme and pray for recombination to occur. And it will, pretty much. You'll have some plasmids that don't recombine with restriction fragments and end up reannealing with themselves but don't worry about that, we've got a way to identify those.
Next you mix your plasmid vectors that hopefully carry a restriction fragment with bacterial cells that are competent to take up foreign DNA, maybe zap 'em with some chemicals or electric current to help the process along, and pray for transformation. Of course, like we said earlier, nothing is 100% efficient, so you'll have some cells that don't take up the plasmid (and some cells will take up plasmids that reannealed without recombining with a restriction fragment) but don't worry about that, we've got a way to deal with those.
Now here's the screening/selection process: You plate the cells out on culture media that has ampicillin and something called X-gal in it. Cells that didn't take up the plasmid will die from ampicillin poisoning, so you can figure anything that grows has the plasmid.
OK, what about cells that took up plasmids that didn't have restriction fragments? Those cells are ampicillin resistant and produce ß-galactosidase. This is where the X-gal comes in.
X-gal is a chromogenic analog of lactose (5-bromo-4chloro-3-indoyl-ß –galactopyranoside) which produces a blue color when broken down by ß-galactosidase. If a cell has a plasmid without a restriction fragment the cell makes ß-galactosidase, which breaks down X-gal and causes the colonies to be blue.
If a cell has a recombinant plasmid (with a restriction fragment inserted in the middle of the lacZ gene) it can't make ß-galactosidase, so it can't break down X-gal, and colonies are white.
So clones containing foreign DNA are identifiable because the colonies they give rise to are white. The individual colonies can be tested for the desired gene product or, if you have a probe, identified by colony hybridization.
