What is a Mutation?
Before we can discuss spontaneous self-correction of gene mutations, we must first understand what a mutation is. Not all of them are bad. In scientifically precise terms, a "mutation" is any change from one hereditary state to another, and a "gene mutation" is any change to the genetic code. (There are also "chromosome mutations.") Each slightly different version of the same gene is called an allele, and there can be many alleles that are equally good to have.
Genes are encoded as strings of nucleotides in a molecule of DNA. There are four nucleotides - adenine, thymine, cytosine, and guanine. A DNA molecule is two strands of nucleotides paired to each other (adenine to thymine, cytosine to guanine), arranged in a double helix. When it's being read, one strand contains the code for the gene and the other is uninvolved.
The nucleotides code for strings of amino acids in a polypeptide chain that, when properly folded into a 3D structure, becomes a protein (or part of a protein - some are multiple polypeptide chains folded together). There are 20 amino acids. Three nucleotides code for one amino acid, and most amino acids are coded for by more than one nucleotide triplet.