What is the difference between "mieux", "meilleur(s)" and "meilleure(s)"?

Mieux is the superlative form of the adverb bien, meaning well.
Meilleur is the superlative form of the adjective bon, meaning good.
Mieux never changes form while meilleur, being an adjective, changes form according to the noun it refers to: meilleure (femine singular), meilleur (masculine plural), meilleurs (masculine plural), meilleures (feminine plural).