Background
Moai are probably the most prevalent of the Gradius traditions, only not appearing in Gradius V, but only because others were saying Gradius V needed new things, not the "traditions" that make Gradius, well, Gradius!
Attack Patterns
The Moai usually stand up or lie flat on the ground, which consists of floating platforms. The Moai always fire Ion Rings, and the rings can be destroyed, but the Moai will simply create more until destroyed. Moai have a moderate amount of health, and can only be killed by shooting them in the mouth. In most Gradius games after II, the Moai will turn red, become more aggressive, and be able to fire many more rings, in addition to being able to stand up and turn around.
Moai Types
Normal Moai: The normal, unaggressive Moai, who only fire energy rings and don't turn around. First appeared in Gradius.
Red Moai: The harder version of the Normal Moai, the Red Moai fire energy rings at a faster pace, and can turn around and stand up. First appeared in Gradius II.
Mechanical Moai: Appearing only in Nemesis 3, mechanical maoi are similar to red maoi, but are harder to destroy. A second type of Mech Moai lays flat on the ceiling or floor and fires an expanding laser similar to the Up / Down lasers.
Jumping Moai: Jumping Moai that can spit rings through themselves, first appeared in Salamander, NOT Gradius II.
Giant Flying Moai: First appeared in Gradius III, these are indestructible flying Moai that spit out smaller Moai. Since they're so large, it can be hard to avoid them. Only appeared in Gradius III.
Monolith Moai: 4 Moai embedded in a rotating monolith that each fire at you. You can only destroy them if you destroy all the Moai. First appeared in Gradius III and went on to appear in Gradius Gaiden.
Inflation Moai: Moai that inflate when shot at, making it hard to focus on the boss you're fighting at the time, and also making it harder to maneuver. Only appeared in Gradius III.
Giant Moai: Giant (and pixelated) Moai that spit out smaller Moai, who in turn fire their rings. Only appeared in Gradius Gaiden.
Mini Moai: The tiny Moai spit out by the Giant Moai, they fire tiny rings, and would go on to appear in Gradius Galaxies as well. First appeared in Gradius Gaiden.
Beam Moai: Blue Moai that not only fire ring AND turn around, but they also fire a huge beam, which still fires as they're crumbling. Only appeared in Gradius Gaiden.
Inert Moai: Black Moai who have no purpose except to be destroyed by the Metal Serpent. The Moai's debris can kill you if you're not careful. Only appeared in Gradius Gaiden.
Blue Moai: Regenerating Blue Moai who act like normal Moai, except they regenerate. Only appeared in Gradius IV.
Black Moai: Regenerating Black Moai that act like Red Moai, except the regenerate. Only appeared in Gradius IV.
White Moai: Moai that help Aa and Mu to defeat you. Only appeared in Gradius IV.
Regenerating Red Moai: Just what the name says, these are Regenerating Red Moai. Only appeared in Gradius Galaxies.
Bubble Moai: Huge Moai that spit bubbles filled with Mini Moai. Only appears in Gradius Galaxies.
Whew! That's a lot of types!
The Moai was also the Gradius icon in Cosmic Wars.
It was also a character in Wai Wai World.