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Since protein-coding DNA is divided into codons three bases long, insertions and deletions can alter a gene so that its message is no longer correctly parsed. These changes are called frameshifts.
For example, consider the sentence, "The fat cat sat." Each word represents a codon. If we delete the first letter and parse the sentence in the same way, it doesn't make sense.
In frameshifts, a similar error occurs at the DNA level, causing the codons to be parsed incorrectly. This usually generates proteins that are as useless as "hef atc ats at" is uninformative.
There are other types of mutations as well, but this short list should give you an idea of the possibilities.