A magical form of fire
References from the canon
Snape used purple fire for his spell to protect the Philosopher's Stone (PS16). Hermione was able to go through it to get back to Ron, by drinking the correct potion.
At the campsite of the Quidditch World Cup, three African wizards were roasting a rabbit on bright purple fire (GF7).
In the wizarding world, purple stands for nobility, having a link to both royalty and religion (Pm).
The magical properties of the fire are unknown; it certainly seems to be hot, as the African wizards used it to roast an animal on it.