"Degenerate" refers to redundancy in the genetic code. Amino acids, the building blocks of proteins, are encoded by codons of three nucleotide bases. Some amino acids are encoded by more than one codon, for example glutamic acid (GAA and GAG). Degeneracy occurs because there are more codons than amino acids. If codons were composed of two bases, this means there would only be 16 amino acids encoded (4^2 = 16). However, there are 21 amino acids and so codons are formed by three bases (4^3 = 64), meaning there must be some redundancy in the code.
Degeneracy means a mutation altering one base in a codon is unlikely to alter the amino acid structure of the encoded protein, because the codon is likely to still encode the same amino acid. This makes the genetic code more fault-tolerant to point mutations.