The Moon Zone is the extra stage in Sonic Advance. It is accessed by obtaining all seven Chaos Emeralds from the Special Stages scattered throughout the game, completing the game as all 4 characters, and then clearing the X Zone while playing as Sonic. It consists of Super Sonic racing along the ground and attacking Eggman, who is in a giant mech (called the "Super Egg Machine" in the official Prima Strategy Guide).
You start with 50 rings, and lose one a second as you go. Eggman will start off guarding the head of the mech (which is the weak point) with spiked claws. The mech will then swing both arms down in an attempt to swat Super Sonic away, and then scrape them along the ground to throw out debris that will knock Super Sonic back if it hits him. While the debris flies, Eggman will fire several laser sweeps that will cause a shockwave of explosions along the ground, which will (you guessed it) also knock Super Sonic back, wasting precious time & rings. To beat him, you need to dodge all of this, and hit the head of the mech to damage it. Eight hits and it's done for (which is much easier than it sounds).
There are NO rings around to pick up to replenish your supply, but you needn't worry. During the fight, Eggman is constantly launching Pac-Man-esque spheres at Super Sonic in an attempt to trap him. If the sphere grabs you, it will hold you for a few moments before it breaks, leaving you open to attack. However, if you manage to dash into the sphere before it can trap you, it will bust open & release some rings for you to collect, replenishing your supply.