Neuter puer avunculi domi manere vult; ambo ad forum ire malunt.

Here neuter means neither (of two). It is not talking about grammatical gender in this sentence.

So neuter puer means neither boy.

This is a very common point of confusion because the English word neuter is familiar from grammar, but in Latin neuter can also be an adjective/pronoun meaning neither. In this sentence it is masculine singular nominative, agreeing with puer.

Because Latin treats neither boy as singular.

So:

• neuter puer ... vult = neither boy ... wants

Even though two boys are being considered, the idea is not one and not the other, so Latin uses a singular noun and a singular verb.

That is why you get:

• neuter puer = neither boy
• vult = wants (singular)

Avunculi is most naturally genitive singular of avunculus, meaning of the uncle.

So:

• puer avunculi = the uncle's boy
• more naturally in English here: one of the uncle's boys

With neuter, the whole phrase comes out as something like:

• neuter puer avunculi = neither of the uncle's boys

A learner may notice that avunculi could also look like a nominative plural form, but here the sense and structure show that it is genitive singular.

Because domi is a locative form meaning at home.

Latin has a small group of words for places that can use a special form instead of a preposition. Domus is one of them.

So:

• domi = at home
• domi manere = to stay at home

English needs at, but Latin does not here.

Because the two phrases express different ideas:

• domi = location in a place: at home
• ad forum = motion toward a place: to the forum

Latin often uses:

• locative for certain place words like domi
• ad + accusative for motion toward somewhere

So:

• domi manere = stay at home
• ad forum ire = go to the forum

Because they depend on verbs of wanting or preferring.

In Latin, verbs like velle (to want) and malle (to prefer) are commonly followed by a complementary infinitive.

So:

• vult manere = wants to stay
• malunt ire = prefer to go

This is very normal Latin syntax.

Malunt is the third person plural of malle, meaning to prefer.

So:

• malunt = they prefer

It looks irregular because malle is an irregular verb. A learner often has to memorize these forms rather than trying to predict them from a regular pattern.

In this sentence:

• ambo ... malunt = both ... prefer

Ambo means both, specifically both of two.

That is why it is plural: it refers to the two boys together.

So the sentence moves like this:

• neuter puer ... vult = neither boy wants ...
  → singular, looking at each one as not the one
• ambo ... malunt = both prefer ...
  → plural, looking at the two together

This contrast is perfectly natural in Latin.

The noun is understood from the context.

Since the sentence has already mentioned puer, ambo means:

• ambo (pueri) = both boys

Latin often leaves out a noun when it is obvious.

Yes. They match the idea of two people very neatly:

• neuter = neither of two
• ambo = both

So the sentence is built around a pair:

• neither boy wants to stay at home
• both prefer to go to the forum

This is a very Latin way to express contrast within a set of two.

Latin word order is flexible, but the order here helps emphasize the contrast.

The sentence begins with Neuter and then later gives ambo, so the reader hears the contrast early:

• neither ... ; both ...

A more mechanically arranged version might be:

• Neuter puer avunculi vult domi manere; ambo ad forum ire malunt.

But the given order is perfectly good Latin and sounds natural.

The semicolon simply separates two closely connected clauses.

The contrast is already clear from the meanings of neuter and ambo, so Latin does not need an extra word like sed (but), though a conjunction could be used in a different version.

So the punctuation helps show:

• first clause: what neither boy wants
• second clause: what both prefer

It is mainly a matter of presentation rather than grammar.