Dr. Mario

Dr. Mario is the first boss encountered in Super Luigi Country. He is based off the spin-off Mario game made by Nintendo of the same name. You encounter Dr. Mario after beating Levels 1-5.

Story
Dr. Mario and the most of the bosses don't really have anything to do with the story other than they're working with Bowser to stop Luigi. However, it is strange that Luigi's own brother is working to stop him, even though it's never mentioned in-game, it is still odd...

Once you beat Dr. Mario, Cranky Kong will say this: "Hey there, wippersnapper! Good job on taking down that digital doctor!"

Dr. Mario returns later in the game in Level 9 as the enemy, Plane Mario and again at the very end of the game in Cranky's Video Game Heroes in 2nd place for some reason.

Boss Fight
Technically, you don't fight Dr. Mario. What you have to do is survive getting hit by his pills for 30 seconds. If you get hit, the fight resets. There is an exploit you can do to make this boss easy, stand near the far left of the capsule and you won't get hit at all.

Once you last the 30 seconds, there will be a barrel that appears, jump in it and continue on.

Trivia

 * The password you get after defeating Dr. Mario is  'doctor'
 * Cranky sort-of breaks the 4th Wall by referring to Dr. Mario as a  'digital doctor', implying he's from a video game.