Famine is a member of the Four Horsemen that you meet along your journey, and the second one that you face in game.
He is located in the middle of a massive desert of his own creation to the north of Mauville city. In order to access this area, you need to use the Hidden Move HM06 Rock Smash.
It is in Famine's Desert that your brother Landon was said to be last seen, and so you head out to investigate.
After heading north to search the desert, you meet Famine, guarded by two Former Chefs and with Alicia at his side.
Right before you reach him, you have another flashback, about when the Horsemen attacked Littleroot town. The memory consists of the houses collapsing on top of your character, and then a blackout.
When you speak to Famine, he compliments your strength at defeating his sister, and states that he believes he has met you before. This familiarity further confuses your character, resulting in a frustrated outburst.
Famine chuckles and confirms that your brother had been there, and fought against his monster, Aerodactyl, before he captured Landon and May. You challenge him, but he leaves before you can. Alicia takes your challenge instead.
The next and final time you see Famine in the ROM is after you enter Meteor Falls under a farce by Alicia. He is with two of the other Horsemen, Pestilence and War (Alicia), and helps them to explain how they exploited your memory loss to their advantage.
After beating War, the entity of Death appears, and banishes all three of them to the Netherworld. This is the last time you see Famine in the ROM.