This question refers to a story told by the players in Shakespeare's Hamlet. The story comes from the second book of Vergil's Aeneid and deals with the fall of Troy.
In Aeneid 2, the Greeks use the stratagem of the wooden horse to enter Troy. In the middle of the night, after the Trojans are asleep, selected Greeks emerge from their wooden hiding place and begin to lay waste to Troy.
One of these Greeks, Pyrrhus (also known as Neoptolemus), who is the son of Achilles, ends up killing Priam, the king of Troy, while Priam's wife Hecuba looks on in horror.
The reason for Hecuba's head covering is twofold, one practical, the other symbolic. From a practical standpoint, Hecuba has a simple head covering because she has been asleep (it is the middle of the night, after all) and it was not uncommon for women to wear some sort of head covering when they went to bed (probably for warmth).
Another reason for Hecuba's head covering is a symbolic one, which Shakespear hints at in his remark "Where late the diadem stood". The removal of the diadem from Hecuba's head distances her from her regal status as Queen of Troy and makes her appear more common. Indeed, after Troy fell, Hecuba would become common because she was designated to become a servant of the Greek warrior Ulysseus (Odysseus).