Learn Russian: Мы хотели бы двухместный номер, пожалуйста.

Мы хотим means 'we want,' which can sound a bit direct or demanding. Adding the particle бы changes it to 'we would like.' This creates the conditional mood, which is the standard, polite way to make requests in Russian.

Why is хотели in the past tense if we are asking for a room right now?

To make this polite 'would like' form (the conditional mood) in Russian, you always use the past tense of the verb plus the particle бы. Since the subject is мы (we), you use the plural past tense хотели. It does not mean the wanting happened in the past; that is simply how this grammatical form is built.

What case is двухместный номер in?

It is in the Accusative case because it is the direct object—the thing that you would like. Because номер is a masculine, inanimate (non-living) noun, its Accusative form looks exactly the same as its Nominative (dictionary) form.

I thought the word for a room was комната. Why is номер used here?

Комната is a room in a house or apartment. When you are talking about a hotel room, you always use номер (which also translates as 'number').

How is the word двухместный built?