Thestrals are huge winged horses with white shining eyes, dragonish faces and necks, and skeletal black bodies. They are attracted to the smell of blood.
Thestrals are invisible to anyone who has not seen death, which means that few of the students in Hagrid's Care of Magical Creatures classes could see them during Harry's fifth year. People used to think they were bad omens, that seeing them meant bad luck, but this is just superstition.
Thestrals have an amazing sense of direction and move magically fast through the air. Dumbledore used them to travel if he didn't care to Apparate (which probably explains how he got to the Ministry (PS16)).
The herd at Hogwarts started with a male and five females. They have lived in the Forbidden Forest for years, and it is believed that the founding members of the herd came to Hogwarts as means of transportation for some students before the introduction of the Hogwarts Express (Pm). A number of thestrals have been born since, beginning with one named Tenebrus, who is a special favorite of Hagrid's. They are used to pull the school carriages from the train station up to the school.
Harry and a group of students flew thestrals from Hogwarts to the Ministry of Magic in an attempt to rescue Sirius. Thestrals were also used by Kingsley, Hermione, Bill, and Fleur in the Battle of the Seven Potters.
In 2017 when Albus Potter went to Hogwarts for the first time, his brother James tells him to watch out for the thestrals, thoroughly scaring his brother.
People often wonder why Harry could not see the thestrals until The Order of the Phoenix, even though he saw his parents die as a baby. J.K. Rowling answers this question on Pottermore:
"Being able to see Thestrals is a sign that the beholder has witnessed death, and gained an emotional understanding of what death means. It is unsurprising that it took a long time for their significance to be properly understood, because the precise moment when such knowledge dawns varies greatly from person to person. Harry Potter was unable to see Thestrals for years after his mother was killed in front of him, because he was barely out of babyhood when the murder happened, and he had been unable to comprehend his own loss. Even after the death of Cedric Diggory, weeks elapsed before the full import of death's finality was borne upon him. Only at this point did the Thestrals that pull the carriages from Hogsmeade Station to Hogwarts castle become visible to him. On the other hand, Luna Lovegood, who lost her own mother when she was young, saw Thestrals very soon afterwards because she is intuitive, spiritual and unafraid of the afterlife."
The new information on the Hogwarts thestrals comes from the Pottermore ebooks.