The general name for an enzyme that does this is a kinase. Technically, the donor molecule can be any high-energy compound, but in practice it's almost always ATP. The recipient molecule can be any biological molecule. (The commonest recipients are proteins, and so the subset of kinases that act on proteins are sometimes called protein kinases.) The donor molecule is thus phosphorylated by the kinase.
It may help to mention a few similar-sounding words, and to show how they differ:
- A phosphatase removes a phosphate group from a molecule.
- An ATPase is an enzyme that catalyses the hydrolysis of ATP to form ADP and Pi.
- A phosphorylase catalyses the breakdown of glucose polymers, like starch, yielding glucose-1-phosphate. (The 'phosphate' bit comes in because some of the bonds are broken with the help of phosphate.)