There are many, many enzymes in the human body that are designed for repairing specific types of DNA damage. Damage to DNA can come in many different forms. UV radiation can cause base pairs to dimerize, or form extra covalent bonds that shouldn't be formed. DNA bases can also become oxidized, methylated, or oxidized. Other chemicals can bind with the DNA or a strand can actually break.
Just as there are different forms of DNA damage, there are different methods of repair. Direct reversal repairs focus on damage that is limited to the base of the nucleotide. Dimers and methylation of the DNA bases can be repaired in this manner. Some different enzymes involved include phytolases (which are actually activated by the same UV light that caused the damage) and methyl guanine methyl transferase which, as the name implies, transfers the extra methyl group off of the DNA base.
Damage to only one of the strands of DNA that cannot be modified usually requires base excision repair. This is where the damaged base is physically removed by a DNA glycosylase enzyme. A DNA polymerase fills in the missing portion and a DNA ligase seals it all together. If the entire nucleotide is damaged, nucleotide excision repair removes and replaces the entire nucleotide.
Breakage of both strands of the double helix is the most serious type of damage. The repair mechanisms for this type of damage are complicated and require a myriad of different enzymes.