Nucleases, also called nucleodepolymerase or polynucleotidase are enzymes able to cut off the phosphodiester bonds that join the adjacent nucleotides in a nucleic acid like RNA and DNA.
Nucleases are highly essential cellular machinery that is naturally present in biological components containing Nucleic acids. In most unicellular organisms, nucleases serve as defense systems for chopping off intruder DNAs. They, however, protect their own DNAs from such damages by means of chemical modifications (for eg Methylation, Acetylation).
Img: Methylated DNA, unable to be degraded by nucleases
They can induce both double and single-strand breaks (nicks) in the polynucleotide chains which is also a critical factor in DNA repair systems. Without inducing a break, a mutated or damaged nucleotide(s) cannot be modified with the normal ones.
Nucleases are classified by the Enzyme Commission as Esterases with the EC-numbers 3.1.11. (EC 3.1.11)
Nucleases are also biological tools used in the field of genetic engineering and molecular cloning. The enzymes that are outsourced naturally, from bacteria or bacteriophages, and artificially by recombinant DNA technology are commercially available for various purposes like the manipulation of DNA and RNA.
On the basis of the location in the DNA chain at which the nucleotides are degraded, Nucleases are primarily of two types:
Exonuclease (from the terminal ends) and Endonuclease (anywhere between the terminal ends)

References:

Nishino, Tatsuya; Morikawa, Kosuke (2002). \”Structure and function of nucleases in DNA repair: shape, grip, and blade of the DNA scissors\” (PDF). Oncogene. Nature Publishing Group. 21 (58): 9022–9032.
RittiĆ©, Laure; Perbal, Bernard (2008). \”Enzymes used in molecular biology: a useful guide\”. Journal of Cell Communication and Signaling. 2 (1): 25–45.
Kaushansky, Kenneth. \”Glossary of Molecular Biology Terminology.\” ASH Education Program Book 2000.1 (2000): 438-456.