This term refers to fact that in most cases in the genetic code used by all life on Earth, one particular amino acid is specified by more than one three-base combination of the four nitrogenous bases (called a codon). There are enough different codons to specify 64 different amino acids, but there are in actuality only 20 amino acids (and three stop codons) used in the making of proteins. Thus, part of the genetic code is degenerate.
From the BioTech Dictionary at http://biotech.icmb.utexas.edu/. For further information see the BioTech homenode.