Learn Russian: Мы приглашаем актёра в театр. - We are inviting the actor to the theater.

Because the actor is the direct object (the person being invited), the noun must be in the Accusative case. Since актёр is a masculine noun and is animate (a living person), it takes the -а ending in the Accusative.

The word театр is also masculine. Why didn't it change to театра?

Театр is also in the Accusative case here because it follows the preposition в to show destination. However, the rule for masculine nouns in the Accusative case depends on whether they are alive. Since a theater is inanimate (not alive), its Accusative form stays exactly the same as its dictionary form.

Why do we use в for "to the theater" instead of на?

In Russian, going to a physical building or enclosed space usually requires the preposition в. You would use на if you were going to an event, like на балет (to the ballet), but the theater itself is a building and takes в.

How does the verb приглашаем match the subject мы?

The dictionary form of the verb is приглашать (to invite). To match the pronoun мы (we), you drop the -ть ending and add the standard first-person plural ending -ем, making it приглашаем.