Why some comparatives are irregular
French comparatives normally put plus in front of the adjective. Three very common words have their own form instead, just as English says better, not more good. English has one word, better, for two French ones: meilleur and mieux.
These forms never take plus: not plus bon and not plus bien.
bon → meilleur
meilleur describes a noun and agrees like any adjective: meilleur, meilleure, meilleurs, meilleures. With the article it is also the superlative: le meilleur, la meilleure (the best).
Ce vin est meilleur que l'autre.
This wine is better than the other one.
Ces fraises sont meilleures que les autres.
These strawberries are better than the others.
C'est la meilleure pizza de la ville.
It's the best pizza in the city.
bien → mieux
mieux says how something is done and goes with a verb. As an adverb it never changes. With le it is the superlative: le mieux (the best).
A quick test: if good fits in English (the coffee is good), use meilleur. If well fits (she sings well), use mieux.
Elle chante mieux que moi.
She sings better than me.
Je me sens mieux aujourd'hui.
I feel better today.
C'est lui qui joue le mieux.
He's the one who plays the best.
Only for “better”
meilleur and mieux exist only for better. For less good and as good, French keeps bon and bien: moins bon, aussi bon, moins bien, aussi bien.
Ce vin est moins bon que l'autre.
This wine is less good than the other one.
Je chante aussi bien que toi.
I sing as well as you.
mauvais → pire
pire (worse) is an adjective with a single singular form for masculine and feminine: le pire jour, la pire idée. The plural adds an s: les pires idées. With the article it is also the superlative: le pire (the worst).
Le temps est pire qu'hier.
The weather is worse than yesterday.
C'est la pire idée !
That's the worst idea!