"Asp" is the modern Anglicisation of the word "aspis," which in antiquity referred to any one of several venomous snake species found in the Nile region. It is believed that aspis referred in Egyptian mythology to what is now known as the Egyptian cobra.
Asp (reptile) Wikipedia
Throughout dynastic and Roman Egypt, the asp was a symbol of royalty. Moreover, in both Egypt and Greece, its potent venom made it useful as a means of execution for criminals who were thought deserving of a more dignified death than that of typical executions. In some stories of Perseus, after killing Medusa, the hero used winged boots to transport her head to Mount Olympus. As he was flying over Egypt some of her blood fell to the ground, which transformed into asps and amphisbaenae.
According to Plutarch (quoted by Ussher), Cleopatra tested various deadly poisons on condemned persons and concluded that the bite of the asp (from aspis - Egyptian cobra, not European asp) was the least terrible way to die; the venom brought sleepiness and heaviness without spasms of pain. The asp is perhaps most famous for its alleged role in Cleopatra's suicide after Mark Antony (her husband) killed himself by falling on his sword due to a false report of Cleopatra killing herself. Some believe it to have been a horned viper, though in 2010, German historian Christoph Schaefer and toxicologist Dietrich Mebs, after extensive study into the event, came to the conclusion that rather than enticing a venomous animal to bite her, Cleopatra actually used a mixture of hemlock, wolfsbane and opium to end her life.
Nonetheless, the image of suicide-by-asp has become inextricably connected with Cleopatra, as immortalized by William Shakespeare:
With thy sharp teeth this knot intrinsicate
Of life at once untie: poor venomous fool
Be angry, and dispatch.
—Cleopatra, Act V, scene II
Antony and Cleopatra
Othello also famously compares his hatred for Desdemona as being full of "aspics' tongues" in Shakespeare's play Othello. (Act 3, scene iii)