Learn Russian: Вечером у нас ужин. - In the evening we have dinner.

Вечером is a special form of the word вечер (evening). In Russian, to say 'in the morning', 'during the day', 'in the evening', or 'at night', you change the ending of the noun rather than using a preposition like 'in'. This form is called the instrumental case.

Why is the word есть missing from the у нас construction here?

You usually see есть in phrases like У нас есть собака (We have a dog) to show ownership. However, for scheduled events, meals, or abstract concepts, есть is dropped. So, у нас ужин is the natural way to say 'we are having dinner' without implying that you own the dinner.

Since this lesson is about the past tense, how would I say 'In the evening we had dinner'?

To put this exact structure into the past tense, you add the past tense of the verb 'to be' (быть). Because ужин (dinner) is a masculine noun, you use был: Вечером у нас был ужин. Alternatively, you could use the regular past tense verb for eating dinner: Вечером мы ужинали.

As a quick refresher, why do we use нас instead of мы?

Нас is the genitive case form of мы (we). The preposition у (by/at) always requires the genitive case. This creates the classic Russian 'to have' structure: у + genitive pronoun + the thing you have.