What are the different moons?

[Mass of BEES]

Are they radically different for each character, if not could somebody give me the basics of the three?

[TheMaster_Rahl]

There are a lot of differences between the moons. In general, the different moons completely change the chars move set (different normals/specials/move properties). As well as changing the meter mechanics, and some defensive options.
There should be a video about the moon systems in the pinned thread Novril put up about tutorials.

I'll highlight some of the key differences here though:

C-Moon is the most like the MBAC version of chars.
-Gets 300% meter, can activate Heat (like a wake up burst) with 100%+, gains red life over time while in Heat or Blood Heat, retains 200% after MAX meter ends unless Arc Drive is used.
-Can Reverse Beat Normal chains (cancel from heavy to medium to light).
-Can Shield (parry) or EX Shield by holding or tapping D, which can be canceled into specials, or normals for EX shield.

H-Moon
-Gets 200% meter, cannot activate Heat on command and instead enters auto Heat at 200%, gains red life over time while in Heat, meter returns to 0% after heat.
-Can Reverse Beat like C-Moon.
-Can only tap Shield, which cannot be canceled, but instead has an auto shield counter.
-Has a 5A6AA series of command normals.
-Has a good shield bunker (alpha counter) that costs 100% meter.

F-Moon
-Gets 300% meter, can only activate Blood Heat in MAX, instantly regains red life on activating Blood Heat or by using Initiative Heat (IH, this is a roman cancel), can use IH while in MAX to get into Heat by tapping 5A+B+C to cancel the recovery of a wide range of things, retains 100% meter after MAX ends unless Arc Drive is used.
-Can not Reverse Beat normals, and has a more strict (light to med to heavy) chain system.
-Can tap or hold shield, which can be canceled into specials or a manual shield counter (236D follow up).
-Can not Dodge.
-Can not Air Dash cancel j.B and j.C normals.

There are more differences, but I think I hit all the main ones.

Edit: Understanding the moon system, and its impact on the game is the most complex part of melty.

[wizard_[ttv]]

Three Moons for every character;

Cresent: Most meter. Can dodge, ex shield, ex guard, burst, automatic heat at 300% meter, manual heat mode. The more balanced of the moons as it has access to everything.

Full: Similar to other anime games like GG/BB, does more damage at the cost of comboability, meter goes to 300%, cannot dodge, ex shield or reverse beat.

Half: General suited for rushing down, shield is more like command parrying, cannot do certain features such as manual heat and ex guard.

Characters general playstyle stays consistent across all the moons but combos and damage is different.

ref: http://wiki.mizuumi.net/w/Melty_Blood/