In principle, meilleur is the comparative/superaltive form of bon, while mieux is the comparative/superlative of bien, the adverb. So mieux generally means "better" in the sense of "in a better way", "more satisfactorily etc". Compare:
c'est un bon ami he's a good friend c'est mon meilleur ami he's my best friend un bon fromage a good cheese un meilleur fromage a better cheese le meilleur fromage (du monde) the best cheese (in the world)
with the use as an adverb, where the literal meaning is "in a better way"/"in the best way" (or something similar):
il chante bien he sings well (="he's a good singer") il chante mieux que moi he sings better than me (="he's a better singer than me") c'est lui qui chante le mieux he's the one who sings best (="he's the best singer")
Now in reality, as you've probably come across, there are some cases where bien is effectively used as an adjective, especially when it means things like "OK", "acceptable", "competent", "good-looking", "comfotrable", "in good health":
C'est bien si...
It's OK if... On est bien là.
We're OK/fine/confortable here.
hence in corresponding cases, mieux also can appear to behave as an adjective:
C'est mieux si...
It's better if... On est mieux ici (que...)
We're better/more comfortable here (than...)
There are also some uses of mieux that you should regard as "set expressions", such as:
il vaut/vaudrait mieux...
it's/it'd be better to..., you'd better... "mieux vaut tard que jamais"
"better late than never" il est/c'est mieux de...+infinitive
it's better to...
So as you go along reading/listening to French, try to look out for these "set expressions".