After you defeat Magikoopa and buy new armor from Croco, you continue to the throne room, where you battle Boomer.
When Boomer is red, use Special attacks on him. When Boomer is blue, use regular attacks on him. He can do a sword attack that hits a single character. He can cast Shaker, which instantly defeats the character that it hits, unless you successfully block the attack (press the button a moment after his sword slash disappears), which will leave that character with 1 HP. He can cast Storm, which damages a single character. Boomer can cast Skewer, a high damage attack on a single character.
After the battle, there will be a cutscene, and you will finally face the giant sword, Exor.