Irregular comparatives: 'meilleur', 'mieux', and 'pire'
In English, we do not say "more good" or "more well". Instead, we use the irregular word "better". French works exactly the same way. While most adjectives and adverbs form their comparative by simply adding plus (more) in front of the base word, a handful of very common words have their own special irregular forms.
Because words like meilleur, mieux, and pire already contain the meaning of "more", you must never put plus in front of them. Saying plus meilleur in French is just like saying "more better" in English: a noticeable grammatical mistake.
The trickiest part of irregular comparatives for English speakers is that English uses the word "better" for both adjectives and adverbs. French makes a strict distinction between meilleur and mieux.
When you want to say that a noun (a person, place, or thing) is better, you use meilleur. This is the irregular comparative of the adjective bon (good). Forms like plus bon simply do not exist. Because it is an adjective, meilleur must agree in gender and number with the noun it describes. For a feminine noun, you will use meilleure.
On the other hand, when you want to say that an action is performed better, you use mieux. This is the irregular comparative of the adverb bien (well). You will never see plus bien in French. Since it is an adverb and describes a verb, mieux never changes its spelling.
To say "worse", French offers the irregular adjective pire, which replaces the concept of plus mauvais. Just like meilleur, it describes nouns. Unlike meilleur, you actually have a choice here: it is also perfectly correct to say plus mauvais in everyday French. However, pire is extremely common and is highly preferred when talking about abstract ideas, situations, or outcomes.
Note that when talking about an action being done worse (the opposite of mieux), French simply uses the regular comparative plus mal.
The adjective petit (small) has a completely regular comparative when you are talking about physical size: you simply say plus petit (smaller). However, when you are talking about an abstract degree, importance, or value, French uses the irregular comparative moindre. It translates closest to "lesser" or "smaller" in an abstract, non-physical sense.
The words meilleur, mieux, pire, and moindre are irregular comparative forms that already mean "more" or "less" by themselves. Never add plus before them. When translating "better", the most important rule to remember is to use the adjective meilleur (or meilleure) to describe nouns, and the adverb mieux to describe actions.