Learn French: Ce coiffeur coupe mieux. - This hairdresser cuts better.

Both words translate to 'better', but they do different jobs. Mieux is an adverb, so it describes an action. Here, it describes the verb coupe (cuts better). Meilleur is an adjective, so it describes nouns, like saying un meilleur coiffeur (a better hairdresser).

How would I change this sentence to say this hairdresser cuts 'the best'?

You would simply add le before mieux to make it a superlative: Ce coiffeur coupe le mieux. Because mieux is an adverb, it never changes to match the gender or quantity of the subject, so it always uses le, never la or les.

Why do we use ce instead of cet or cette?

Coiffeur is a masculine noun starting with a consonant, so we use the masculine demonstrative adjective ce (this). If the masculine word started with a vowel sound, like ami, we would use cet. If the hairdresser were female (une coiffeuse), we would use the feminine cette.

Does the verb couper specifically mean cutting hair?