Literary device, originally employed in Ancient Greek drama, consisting of rapidfire alternating dialogue between two characters. Shakespeare, having learned the trick from the Roman playwright Seneca, makes use of it in many of his plays. Stichomythia lends itself nicely to rising dramatic tension, so you'll often see it employed when two characters are exchanging particularly heated words.
Editor Stephen Greenblatt offers a good example in his Norton Shakespeare from A Midsummer Night's Dream, where Shakespeare is using the device to heighten the mutual distress of his lovers.
Lysander: The course of true love never did run smooth,
But either it was different in blood--
Hermia: O cross!--too high to be enthralled so low.
Lysander: Or else misgrafted in respect of years--
Hermia: O spite!--too old to be engaged to young.
Lysander: Or merit stood upon the choice of friends--
Hermia: O hell!--to choose love by another's eyes.