Learn Polish: Kto ją zaprasza do kina? - Who is inviting her to the cinema?

ją is a short pronoun meaning 'her'. In Polish, these short pronouns like to tuck themselves into the middle of the sentence. They usually avoid being at the very end of a sentence, so putting it right before the verb sounds much more natural than saying 'Kto zaprasza ją?'.

Are there other words for 'her' in Polish, or just ją?

For a direct object (the person being invited), we use the Accusative case, which is ją. You might also see nią in other sentences, but that longer form is used after prepositions. Here, since there is no preposition before 'her', the short form ją is exactly what we need.

Why does the word for cinema change to kina here?

The basic dictionary word for cinema is kino. However, the preposition do (meaning 'to') always forces the noun that follows it into the Genitive case. For neuter nouns ending in -o like kino, the Genitive ending changes the -o to an -a, giving us do kina.

Why do we use the 'he/she/it' form of the verb (zaprasza) for the word kto?

Just like in English, where we say 'Who is' instead of 'Who are', the Polish question word kto (who) always acts as a third-person singular subject. That is why we use zaprasza instead of forms like zapraszasz (you invite) or zapraszają (they invite).