Thunder Mario (AKA Lightning Mario) is one of Mario's Power-Ups. Mario can access this form by using a Thunder Flower or a Lightning Rod. It is considered a "companion" ability to Ice Mario & Fire Mario, since they all come from flowers and they have similar abilities.
Thunder Mario can shoot balls of electricity from his hands. They can kill normal enemies in one hit and can paralyze stronger enemies that take multiple hits to kill, giving him an advantage over some enemies and more time to kill them. In New Super Mario Bros. Omega, however, every enemy would be stunned, but weak foes could be killed after one touch when stunned. Even bosses could be stunned, as long as they're not big.
The thunder balls also bounce longer than fireballs. First, they bounce high, but bounces lower and lower after every bounce, until the bounce is so low that it actually "moves on it's own". Then it turns into a small shock and disappears.