Mieux is an adverb, comparative of "bien" (also an adverb).
Meilleur(e) is an adjective, comparative of "bon" (also an adjective)
Both bien and bon translate as "good" in English but there's a difference.
"Cette chaise est mieux que celle là" is not correct. It's commonly used in spoken language because it's shorter but you shouldn't write it down. We tend to use "mieux" especially at the end of the sentence : "Cette chaise est bien mieux" (This chair is much better). You should say "Cette chaise est bien meilleure."
So to sum up :
Cette chaise est meilleure que celle là
(This chair is better than that one : adjective)
(Don't forget to add an "e" when it's feminine ! (une chaise))
Il nage mieux que l'année dernière
(He swims better than last year : adverb)
And as it has been said : always "bien meilleur", and you can say "beaucoup mieux" or "bien mieux".
And of course, "best" is always "meilleur(e)"
EDIT : Acutally Mieux can be used as an adjective. It's used instead of meilleur when there is no noun directly after it :
Il est en meilleure santé
(He's in better health)
Il va mieux.
(He's feeling better/ he's better)
Turns out "mieux" can also be a noun... I'm sorry that French can be so confusing ! But it's the same rule as above : if there is a noun directly after use meilleur, if not use mieux.
Je m'attendais à mieux
(I expected better)
Je m'attendais à un meilleur film
(I expected a better movie)