Hypoxanthine-aminopterin-thymine (HAT) medium is a type of agar growth medium (a selective medium) which is used to select for cells that are able to make their nucleic acids out of hypoxanthine and/or thymine.
All cells are prevented from making nucleotides by any other means (due to the presence of aminopterin or other drugs in the medium), so cells which are unable to use hypoxanthine and/or thymine to do this will not grow, multiply, and become visible colonies. This leaves the cells which are being selected for.
From the BioTech Dictionary at http://biotech.icmb.utexas.edu/. For further information see the BioTech homenode.