Simple definition: parapsychology is the scientific study of "psychic" phenomena such as telepathy, precognition, near-death experiences, and telekinesis, and several other types of anomalous phenomena. However, it is not the study of all paranormal phenomena- parapsychologists don't study such things as UFOs and Bigfoot. The term parapsychology is often abused by people who have no scientific credentials and, far more important, do not use truly scientific methods in their work.

Nuanced definition: according to the Parapsychological Association, parapsychology is the scientific study of certain types of paranormal phenomena, or of phenomena which appear to be paranormal. The term is based on the Greek para (beside/beyond), psyche (soul/mind), and logos (account/explanation). It was coined by psychologist Max Dessoir in or before 1889. Its first appearance was in an article by Dessoir in the June 1889 issue of the German publication Sphinx. J. B. Rhine later popularized the word as a replacement for the earlier term psychical research, during a shift in methodologies which brought laboratory methods to the study of psychic phenomena. In contemporary research, the term parapsychology refers to the study of psi, a general blanket term used by academic parapsychologists to denote anomalous processes or outcomes.

You may have seen definitions of parapsychology like this:

"a field of study concerned with the investigation of evidence for paranormal psychological phenomena (as telepathy, clairvoyance, and psychokinesis)." Merriam-Webster

Such a definition is wrong because no one has proven that parapsychological phenomena are psychological, although most of them do seem to have some connection to psychological states.

You may even have seen definitions like the following:

"The study of the evidence for psychological phenomena, such as telepathy, clairvoyance, and psychokinesis, that are inexplicable by science." yourdictionary.com

This prejudiced definition puts parapsychology utterly outside the realm of science, because if a thing cannot be scientifically explained, there is no use studying it.