"They aren't unlucky, they're dead clever an' useful! Thestrals aren' dangerous! All righ', they might take a bite outta you if yeh really annoy them--"
-- Rubeus Hagrid
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. They are amazingly magical animals, however. People used to think they were bad omens, that seeing them meant bad luck, but this is just superstition (OP21, OP34, FB). 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) and maybe even how Hagrid got to the Hut on the Rock by flying (PS5)).
The herd at Hogwarts started with a male and five females. A number of them have been born since, beginning with one named Tenebrus, which is a special favorite of Hagrid's (OP21). They are used to pull the school carriages from the train station up to the school (OP10). Harry and a group of students flew Thestrals from Hogwarts to the Ministry of Magic in an attempt to rescue Sirius (OP34).