"A ghost, as I trust that you are all aware by now, is the imprint of a departed soul left upon the earth..." - Professor Snape
Ghosts are a spirit that are either afraid of death, or have some kind of strong connection to the location that they haunt. They are visible and appear as a greyish-silver apparition of their living selves. They don't have much physical influence - they can pass through solid objects without damaging themselves or the object. Their presence does cause the temperature to drop in their immediate vicinity.
Hogwarts is home to multiple ghosts, which is the most haunted place in Great Britain. While waiting to be Sorted, Harry saw about twenty ghosts come through the back wall. Here is a little background information on a few of them...
HOUSE GHOSTS
Each of the Hogwarts House has a patron ghost, who once belonged to that House. Their ghost serves as a representative of their House, as well as being a messenger or guide for their House's students.
Gryffindor
Gryffindor's ghost is Sir Nicholas De Mimsy-Porpington, commonly known as "Nearly-Headless Nick" by the students.
Sir Nicholas is a friendly ghost, but a bit of a snob. During his lifetime, he was part of King Henry VII's court and believed himself to be a more accomplished wizard than he really was.
He was executed with a blunt axe, which did not fully cut off his head. This means that he isn't eligible for the Headless Hunt, a group of headless ghosts that Sir Nicholas wants to join.
Hufflepuff
Hufflepuff is haunted by the "Fat Friar" - his real name is unknown. He is a jolly and forgiving man, who in life had dedicated himself to religion.
The Fat Friar was executed after the senior churchmen grew suspicious of his ability to cure the pox by "poking peasants with a stick". He also had a habit of pulling rabbits out of the communion cup.
He is still resentful that he was never made a cardinal. He is often the only one to speak in support of Peeves, frequently suggesting that he should have another chance.
Ravenclaw
Ravenclaw's ghost is Helena Ravenclaw, daughter of Rowena Ravenclaw. Helena is known simply as "The Grey Lady" by most students.
When she was alive, Helena stole Rowena's diadem and ran away to Albania. When Rowena became fatally ill, she asked the Baron to find Helena. He was in love with Helena, but she did not return his affections. Helena refused to come with him and rejected him again. Furious, he killed her.
The students of the other Houses think she is timid and never speaks, but she does talk to the Ravenclaws. She is particularly helpful when they are lost, or have mislaid something.
Slytherin
Slytherin's ghost is known as the "Bloody Baron". He is described as having "blank staring eyes, a gaunt face, and robes stained with silver blood".
In life, he had a volatile temper and was in love with Helena Ravenclaw. He was sent by her mother to bring Helena home. When she refused and rejected his affections, he stabbed her. Overcome with grief and remorse, he killed himself with the same weapon. He carries chains as an act of penitence.
The Bloody Baron is known for being extremely unsocial and most students are afraid of him, including some Slytherins. He is the only ghost who is able to control Peeves.
OTHERS
Myrtle Warren
Myrtle was a muggle-born student in Ravenclaw House. At Hogwarts she was bullied about her glasses and acne, as well as her difficulty making friends. She now haunts the toilet where she died and is known as "Moaning Myrtle".
On 13 June 1943, Myrtle was crying in one of the bathroom stalls. Another student, Olive Hornby, had been making fun of Myrtle's glasses. While she was in there, Myrtle heard a boy speaking in a "different language". She opened the door to tell him to go away.
It was Tom Riddle, speaking Parseltongue to open the Chamber of Secrets. Tom commanded the basilisk to look her in the eye, which killed her instantly. She was Tom's first victim and he used her murder to make his first horcrux: the diary.
Myrtle's body was found hours later by Olive.
HONOURABLE MENTION
Peeves
Peeves is a poltergeist, not a ghost and has been causing chaos at Hogwarts since it was founded. It is thought that he is a tangible manifestation of the students' mischief. He is said to be the most notorious and troublesome poltergeist in British history.
Peeves is a constant pain to the Hogwarts inhabitants, especially their caretakers. The only ones who can control him are Professor Dumbledore and the Bloody Baron. He is, however, loyal and devoted to Hogwarts. He opposed Delores Umbridge and fought against Voldemort during the Battle of Hogwarts.