Stanotte abbiamo ballato fino alle due.

It can mean either, depending on the tense and context.

• With a past tense, as here — Stanotte abbiamo ballato... — it normally means last night.
• With a future or present/future context, it can mean tonight.

So in this sentence, because of abbiamo ballato (past), stanotte is understood as last night.

Because in modern Italian, stanotte is a fixed adverbial expression, written as one word.

This is very common with time expressions such as:

• stamattina = this morning
• stasera = this evening / tonight
• stanotte = tonight / last night

So you should learn stanotte as a single unit.

Abbiamo ballato is the passato prossimo, a very common past tense in Italian.

It is formed with:

• abbiamo = present tense of avere
• ballato = past participle of ballare

So:

• abbiamo ballato = literally we have danced
• but in normal English it is often translated as we danced

In this sentence, the most natural English translation is usually we danced.

Because ballare normally forms the passato prossimo with avere, not essere.

So:

• abbiamo ballato = correct

Italian past tenses use either avere or essere as the auxiliary verb, depending on the main verb. Unfortunately, this is something you often have to learn verb by verb.

For ballare, the standard auxiliary is avere.

Because when the auxiliary is avere, the past participle usually does not agree with the subject.

So even though abbiamo means we, the participle stays in its basic form:

• abbiamo ballato

This is different from many verbs that use essere, where the participle usually agrees with the subject.

For example:

• siamo andati = we went / we have gone
  here andati agrees with we (masculine or mixed group)

But with avere:

• abbiamo ballato = no agreement with the subject

Italian often drops subject pronouns when they are not needed.

The verb ending already tells you the subject:

• abbiamo = we have

So noi is optional.

• Abbiamo ballato = perfectly normal
• Noi abbiamo ballato = also possible, but more emphatic

You would use noi if you want contrast or emphasis, for example:

• Noi abbiamo ballato, loro no.
  We danced, they didn’t.

It means until two o’clock.

Breakdown:

• fino a = until / up to
• le due = two o’clock

When a comes before le, they combine:

• a + le = alle

So:

• fino alle due = until two

Because in Italian, clock times normally use the definite article.

So Italians say, literally:

• at the two
• at the three
• at the five

Examples:

• alle due = at two
• alle tre = at three
• alle cinque = at five

This is why fino a + le due becomes fino alle due.

A useful special case:

• all’una = at one

because una is singular.

Yes, but stanotte is usually the more natural, compact expression for tonight / last night.

• stanotte = very common in everyday Italian
• questa notte = also possible, but often a bit more explicit or marked depending on context

In many ordinary sentences, native speakers would prefer stanotte.

Yes. Italian word order is fairly flexible.

The sentence:

• Stanotte abbiamo ballato fino alle due.

starts with the time expression, which is very natural.

You could also say:

• Abbiamo ballato fino alle due stanotte.

That is grammatical too, but the emphasis shifts slightly. Putting stanotte first makes the time frame feel like the starting point of the sentence.

Because abbiamo ballato presents the action as a completed event.

• Stanotte abbiamo ballato fino alle due.
  = Last night we danced until two.
  This sounds like a finished event with a clear endpoint.

By contrast, ballavamo is the imperfetto, which usually gives background, ongoing action, or repeated action in the past.

For example:

• Stanotte ballavamo quando è arrivato Marco.
  = Last night we were dancing when Marco arrived.

So in your sentence, abbiamo ballato is used because the speaker is talking about a completed action.