We say that the genetic code is degenerate because there are several codons, which code for the same amino acid. For example UCA, UCU, UCG and UCC all code for Serine.
This is due to the fact that mRNA coding for an amino acid only has 4 possible letters A, U, C and G and that there are 20 amino acids that need to be coded. If codons consisted of only one letter we would have 41 =4 possible combinations (not enough combinations to code 20 amino acids) if codons consisted of two letters 42 = 16 (not enough combinations to code 20 amino acids) only when codons consist of three letters we get 43 = 64 combinations and we have enough combinations to code for 20 amino acids. However 64 is a lot more than 20, therefore different 3 letter combinations (codons) are used to code for a single amino acid.