Mod never looks the same way twice, so it's hard to peg him with a physical description. The costume he seems to use most, simply because it's his most recognizable, is that of a lion with a gold crown and a red cape.
His realism, in other words how much he looks like he's obviously a costume, depends on how much his brain or what passes for one is working. Whether or not this is an intentional constraint or not, it's a great mechanism for keeping him in a state of general bliss. The less he thinks, the happier he becomes. Ignorance is bliss, after all. The happier he becomes, the more realistic he becomes. Mod, realistic, is virtually indistinguishable from a regular flesh-and-blood creature. He has no seams, no stitches, his skin is smooth and warm, his fur correctly aligned, his coloration appropriately mottled. The more he thinks, the more he becomes aware of life's imperfections. His lustrous and regal fur turns into monochromatic fun-fur, his limbs become baggy, and his mask has a tendancy for coming unhinged and falling off. If caught in thought-traps, Mod disintegrates into his component parts, revivable by being worn.
Mod seems to be all-powerful at times, but anything he does to others is an illusion and can be dispelled with the slightest thought. Or it can be made real with the slightest thought.
Because he's not tied to a central body, he can be destroyed an infinite amount of times and still stay alive. Like Mel, his essence lies somewhere else.