Learn Russian: У тебя есть ключ от чужой машины? - Do you have the key to someone else's car?

In Russian, keys are thought of as being "from" the thing they open. So you always use the phrase ключ от (key from) followed by the Genitive case, instead of translating "to" directly.

I thought чужой was the masculine dictionary form. Why is it used here with the feminine noun машины?

You are right that чужой is the masculine Nominative form! However, this sentence uses the Genitive case because of the preposition от. The dictionary form of the feminine adjective is чужая, and its feminine Genitive ending becomes -ой. So, чужой here is actually the feminine Genitive form matching машины.

Does машины mean "cars" (plural) in this sentence?

No, here it means "car" in the singular. Because the preposition от requires the Genitive case, the feminine noun машина changes its ending to -ы. This Genitive singular form just happens to look exactly like the Nominative plural form for "cars".

Is чужой an adjective or a possessive pronoun?

Grammatically, чужой is an adjective, but it translates perfectly to the English possessive concept of "someone else's". Just like a normal adjective, its ending changes to match the gender, number, and case of the noun it describes.