A position of a codon is said to be a n-fold degenerate site if only n of four possible nucleotides (A, C, G, T) at this position specify the same amino acid. A nucleotide substitution at a 4-fold degenerate site is always a synonymous mutation with no change on the amino acid.
A less degenerate site would produce a nonsynonymous mutation on some of the substitutions. An example (and the only) 3-fold degenerate site is the third position of an isoleucine codon. AUU, AUC, or AUA all encode isoleucine, but AUG encodes methionine. In computation, this position is often treated as a twofold degenerate site.
A position is said to be non-degenerate if any mutation at this position changes the amino acid. For example, all three positions of methionines AUG are non-degenerate, because the only codon coding for methionine is AUG. The same goes for tryptophans UGG.
There are three amino acids encoded by six different codons: serine, leucine, and arginine. Only two amino acids are specified by a single codon each. One of these is the amino-acid methionine, specified by the codon AUG, which also specifies the start of translation; the other is tryptophan, specified by the codon UGG.