There are several of mechanisms for DNA repair, when it comes to error making. DNA repair process consists of mechanism for the recognition of error and mechanism that repairs that error.
During DNA replication enzymes DNA polymerases check and fix mispaired bases (if there is) in a process called proofreading.
If there is any unfixed pair, it is going to be detected and replaced in a process called mismatch repair after the DNA replication. Complex of proteins cuts the DNA near the mismatch and DNA polymerase then replaces the missing section.
In base excision repair, just the damaged base is removed, while in nucleotide excision repair a patch of nucleotides is removed.
Double-stranded break can also be repaired via non-homologous end joining and homologous recombination.