Alter ad forum ambulat, alter domi manet.

In Latin, alter ... alter means one ... the other or the one ... the other when there are exactly two people or things in view.

So:

• Alter ad forum ambulat = One walks to the forum
• alter domi manet = the other stays at home

This is a very common Latin pattern.

Not quite. Alter usually means the other of two or one of the two. That is different from a more general another.

A learner should especially remember:

• alter = one/the other out of two
• if Latin wants to talk about another in a less specific sense, it may use other words depending on context

So in this sentence, alter is not random: it tells you there are two people being contrasted.

Because ad means to or toward, and it takes the accusative case.

Here:

• ad = to/toward
• forum = accusative singular of forum

So ad forum means to the forum.

This shows motion toward a place.

Forum is in the accusative singular.

That is because the preposition ad always takes the accusative when it means to or toward. So:

• nominative: forum
• accusative: forum

Since forum is a neuter second-declension noun, its nominative and accusative singular look the same, which can be confusing at first.

Domi is a special form meaning at home. It is a locative form.

Latin keeps this old locative case with a few words, especially names of cities, small islands, and certain common place-words such as:

• domi = at home
• ruri = in the countryside
• humi = on the ground

So instead of saying in home, Latin simply says domi.

Because manet means remains/stays, so it describes being in a place, not motion toward it.

Compare:

• domi manet = he/she stays at home
• ad domum ambulat = he/she walks home / to the house

So:

• ad + accusative = motion toward
• domi = location, at home

Since this sentence contrasts movement and staying, Latin uses different constructions very naturally.

Both are present tense, third person singular, active voice.

• ambulat = he/she walks
• manet = he/she stays/remains

Because Latin often leaves the subject unstated, these verb endings are important. The -t ending tells you the subject is he, she, or it.

Latin often does not need separate subject pronouns, because the verb ending already shows the person and number.

Here:

• ambulat = he/she walks
• manet = he/she stays

And alter ... alter supplies the idea of the two people: one ... the other.

So Latin does not need to add is or ille unless it wants extra emphasis.

Either can be correct, depending on context.

The Latin present tense can often correspond to English:

• walks
• is walking

So Alter ad forum ambulat may be understood as:

• One walks to the forum
• One is walking to the forum

Likewise manet can be stays or is staying.

Latin word order is more flexible than English word order. The sentence could still be understood even if the words were arranged differently.

Here, domi manet is a natural order, with the place-word before the verb: at home stays.

The overall sentence is nicely balanced:

• Alter ad forum ambulat
• alter domi manet

That balance helps emphasize the contrast between the two people.

Yes, that is a perfectly natural English translation if the meaning has already been explained.

Strictly speaking, ambulat means walks, not just goes, but English often prefers goes if the exact manner of movement is not important.

So:

• more literal: One walks to the forum, the other stays at home
• more natural in some contexts: One goes to the forum, the other stays at home

Manet means stays, remains, or continues to be in a place or condition. It is stronger than simply is.

So:

• domi est = he/she is at home
• domi manet = he/she stays/remains at home

In this sentence, manet makes a good contrast with ambulat:

• one person is moving
• the other remains where he or she is

Yes. It is very simple, balanced Latin.

The structure is almost symmetrical:

• Alter
  • place + verb
• alter
  • place + verb

This kind of parallel structure is common in teaching sentences because it makes the grammar easy to see and the contrast very clear.