Wir lachen über den Witz.

Because in German, the verb lachen often goes with über when you say what you are laughing about.

• über etwas lachen = to laugh about / at something

So German does not normally say jemanden/etwas lachen in this meaning. You need the preposition:

• Wir lachen über den Witz.

This is just the standard pattern that belongs with lachen in this context.

Because über here takes the accusative case, and Witz is a masculine noun.

The forms are:

• nominative: der Witz
• accusative: den Witz

Since the phrase is über den Witz, the noun has to be in the accusative.

Not always. Über is a two-way preposition (Wechselpräposition), which means it can take either:

• accusative for direction/movement
• dative for location/position

But with lachen über, it is not about physical movement or location. It is a fixed verb-preposition combination, and it uses the accusative:

• über den Witz
• über die Geschichte
• über das Problem

So in this sentence, you should simply learn the pattern über + accusative after lachen.

It is the present tense, 1st person plural:

• ich lache = I laugh
• du lachst = you laugh
• er/sie/es lacht = he/she/it laughs
• wir lachen = we laugh
• ihr lacht = you all laugh
• sie/Sie lachen = they laugh / you laugh (formal)

So Wir lachen means we laugh or we are laughing, depending on context.

In English, laugh at is a verb plus particle/preposition combination. German expresses this idea differently, usually with:

• über etwas lachen

So even though the English translation may be laugh at the joke, German uses über. This is one of those places where you should not translate word-for-word from English.

No, that sounds wrong in standard German.

You need:

• Wir lachen über den Witz.

The verb lachen does not normally take a direct object in this meaning. Instead, it uses the prepositional phrase über + accusative.

Witz is masculine:

• der Witz

That matters because the article changes depending on case:

• nominative: der Witz
• accusative: den Witz
• dative: dem Witz
• genitive: des Witzes

In this sentence, because über goes with the accusative, you get:

• über den Witz

Yes. German word order is flexible, but the conjugated verb usually stays in second position in a main clause.

The neutral order is:

• Wir lachen über den Witz.

But you could also say:

• Über den Witz lachen wir.

That puts more emphasis on über den Witz.

So the sentence can change order, but lachen still has to be in the correct position in a main clause.

This is an important distinction.

• über jemanden/etwas lachen = to laugh about someone/something; sometimes neutral, sometimes slightly negative depending on context
• jemanden auslachen = to laugh at someone in a mocking or cruel way

Examples:

• Wir lachen über den Witz. = We are laughing at/about the joke.
• Sie lachen ihn aus. = They are mocking him / laughing at him.

So auslachen is usually much harsher.

You would say:

• Wir lachen nicht über den Witz.

In German, nicht usually goes before the part you want to negate, and in this sentence it naturally comes before the prepositional phrase:

• Wir lachen nicht über den Witz.

Über is pronounced roughly like OOH-ber, but with the German ü, which does not exist in standard English.

A helpful approximation:

• say ee as in see
• round your lips as if saying oo

That gives you something close to ü.

So:

• über ≈ Ü-ber

Also remember that the b in über sounds like a normal b here, and the stress is on the first syllable: Ü-ber.

Yes. English may translate it in slightly different ways depending on context:

• We are laughing at the joke.
• We are laughing about the joke.

German uses über for the thing that causes the laughter, so both English versions can fit. The exact best translation depends on the situation, but the German structure stays the same:

• über + accusative

Because wir is a personal pronoun, not a noun with an article.

You simply use:

• wir = we

It does not take der/die/das or ein/eine. The only part changing for grammar here is the verb and the noun phrase after über:

• wir lachen
• über den Witz