Learn Portuguese (Portugal): Eles têm o mesmo carro.

mesmo means "same" and it has to agree with the noun it describes. Since carro (car) is a masculine, singular noun, we use o mesmo. If they had the same house (casa), you would say a mesma casa.

I thought adjectives usually go after the noun in Portuguese. Why does mesmo come before carro?

You're right that most descriptive adjectives come after the noun (like carro vermelho for "red car"). However, mesmo is one of the few special adjectives that almost always comes before the noun when it translates to "same".

Can mesmo mean anything else? I feel like I've seen it used differently.

Yes! Besides meaning "same", mesmo is frequently used as an adverb to mean "really" or "exactly" (like É mesmo bom! meaning "It is really good!"). But when you see it paired with a "the" word (o, a, os, as) right before a noun, it means "the same".

Why does the verb têm have a circumflex accent (the little hat) on the 'e'?