Learn Italian: In estate vengono nel nostro albergo. - In summer they come to our hotel.

In Italian, we usually just use in or di before a season without the word for "the" (the definite article). So, "in summer" is simply in estate (or d'estate).

Where is the word for "they" in this sentence?

It is built right into the verb! Vengono means "they come" (from the irregular verb venire). In Italian, you usually drop subject pronouns like loro (they) because the verb's ending already tells you who is doing the action.

Why do we use nel here if the English translation says "to our"? Doesn't nel mean "in the"?

Yes, nel is a combination of in and il (in the). In English we say we are going "to" a hotel, but in Italian, verbs of movement often use the preposition in for specific places like hotels, banks, and mountains.

I learned that possessives need the word for "the" before them, like il nostro. Where did the il go here?

It is hiding inside nel! Because nel is a combination of in and il, the phrase nel nostro albergo literally translates to "in the our hotel". This perfectly follows the Italian rule of putting the article before the possessive.