They are both portrayed by Boris Karloff.
In this film, Dr. Jekyll uses Hyde to exact vengeance upon others who ridicule his scientific research, as shown at the start of the film when he killed one of those: a colleage of his named Dr. Poole.
He is also in love with his niece Vicky and attempts to murder her new love interest, a reporter named Bruce.
Abbott and Costello play British policemen who are kicked off the force after failing to break up a riot at a women's suffrage demonstration.
They realize that capturing Mr. Hyde is the only way they will get their jobs back, but after Costello locks him up, he transforms back into Dr. Jekyll. After Costello drinks a serum in Jekyll's lab that turns him into a giant mouse, they attempt to have Jekyll arrested.
When Abbott, Costello, Bruce and Vicky return to the house, he transforms once more (and accidentally turns Costello as well with the injection that turns him into Hyde) and a chase ensues throughout London.
After a long chase, Jekyll returns to the house and attempts to kill Vicky, when Bruce stops him. Jekyll attempts to escape out from an upstairs window by climbing up using the help of a close tree, but a branch snaps, causing Jekyll/Hyde to fall to his death. As he dies, he transforms from his monster form back into human in front of countless police and civillian witnesses, ultimatly revealing that he was the monster behind all of the murders.
Meanwhile, when Costello (also in the form of Mr. Hyde) is brought into the police station, he too transforms back and is cured, but not before biting four policemen and the inspector, turning them all into monsters as well.
This characterization of Jekyll has gained much negative attention as he is already evil in his own right, unlike other versions of the character, and has no qualms with Hyde committing murder-meaning there is no struggle between the two.