Learn Russian: Добро пожаловать, мы вас очень ждали. - Welcome, we've really been waiting for you.

It roughly translates to "Kindly arrive" or "You are kind to visit." Добро comes from the word for "good" or "kindly," and пожаловать is a formal, older verb meaning "to visit" or "to arrive." Today, it is just a fixed phrase for "Welcome!" that you use to greet someone who has just arrived.

Why use the imperfective verb ждали here instead of a perfective form?

The imperfective ждали focuses on the ongoing process of waiting. It emphasizes that the speakers spent a continuous period of time waiting for the guests right up until their arrival.

I learned that очень means "very," but here it's translated as "really." Why is that?

In English, we don't usually say "we very waited." We say "really" or "waited a lot." But in Russian, очень is commonly used directly with verbs to show a high degree of intensity or emotion, so очень ждали is the natural way to express "we really waited" or "we were looking forward to you a lot."

Which case is вас in this sentence?

It's the accusative case of вы (you). When you wait for a specific person or people, the verb ждать (to wait) takes the accusative case. Because вы refers to living people, its accusative form looks exactly the same as its genitive form (вас).