Мы увидели луну ночью.

Because it is the direct object of the verb увидели (saw).

• Dictionary form: луна = moon
• In this sentence, moon is what was seen, so Russian uses the accusative case
• For a feminine noun ending in -а, the accusative singular usually changes -а to -у

So:

• луна = the moon
• увидели луну = saw the moon

Here ночью means at night / during the night.

This is a very common Russian way to express time: the noun ночь becomes ночью, which is the instrumental singular form.

So:

• ночь = night
• ночью = at night

It functions almost like an adverb in this sentence.

Увидели is the past tense plural of увидеть.

Увидеть is a perfective verb, so it usually means:

• to see
• to catch sight of
• to notice

In this sentence, it suggests a completed event: we saw the moon.

Because the subject is мы = we.

In the Russian past tense, verbs agree with the subject in gender/number:

• увидел = he saw
• увидела = she saw
• увидело = it saw
• увидели = they saw / we saw

Since мы is plural, the verb must be увидели.

This is an aspect question.

• увидели = perfective, focuses on the completed fact of seeing something
• видели = imperfective, often focuses more on the experience, process, or repeated seeing

Compare:

• Мы увидели луну ночью. = We saw the moon that night / We caught sight of the moon at night.
• Мы видели луну ночью. = We were seeing the moon at night / We used to see the moon at night / We saw the moon (with less emphasis on the single completed event)

In many contexts both are possible, but увидели sounds more like we managed to see it / we caught sight of it.

Russian does not have articles like a and the.

So луну can mean:

• a moon in some contexts
• the moon in this context

Usually the meaning is clear from context. Since people normally mean Earth’s moon, English translates it as the moon.

No, Russian word order is fairly flexible.

This sentence is the most neutral order:

• Мы увидели луну ночью.

But other orders are possible, for example:

• Ночью мы увидели луну.
• Луну мы увидели ночью.

These versions can change the focus or emphasis, but the basic meaning stays the same.

Yes, often you can.

Russian often drops subject pronouns when the meaning is clear from the verb form or context.

So both can work:

• Мы увидели луну ночью.
• Увидели луну ночью.

The second version sounds more natural if it is already clear who we are.

Ночью is the normal way to say at night.

• ночью = at night / during the night

By contrast, в ночь usually appears in more specific expressions, for example:

• в ночь на понедельник = on the night leading into Monday

If you want to mean on that night, Russian often says:

• в ту ночь = that night

So in your sentence, ночью is the natural choice.

The stress is:

• мы уви́дели луну́ но́чью

A rough pronunciation guide:

• мы = my
• уви́дели = oo-VEE-dee-lee
• луну́ = loo-NOO
• но́чью = NO-chyu

The чью part in ночью may feel tricky at first. It is approximately chyu, but softer than in English.

It can do either, depending on context.

In this sentence, because увидели is a single completed past action, ночью will normally be understood as:

• during the night
• that night
• at night on that occasion

So the sentence is naturally understood as a specific event, not a general habit.

Grammatically you could say that in English, but it is not the most natural translation.

The most natural English translations are:

• We saw the moon at night.
• We saw the moon that night.
• We saw the moon during the night.

Which one fits best depends on the wider context.