Mater filiam secum ad forum ducit.

Because mater is in the nominative case, which is the case normally used for the subject in Latin.

Also, the verb ducit means she leads / she is leading / she brings, so it matches a third-person singular subject.
So mater = the mother is the one doing the action.

Because filiam is accusative singular, and the accusative case is used for the direct object.

• filia = daughter as a subject
• filiam = daughter as the object

So in this sentence, the mother is acting on the daughter: she is leading her daughter.

Ducit is from ducere, meaning to lead, to guide, or sometimes to bring/take depending on context.

Grammatically, ducit is:

• present tense
• active voice
• indicative mood
• third person singular

So it means:

• she leads
• she is leading
• sometimes naturally in English, she takes or she brings

In this sentence, English often translates it more naturally as The mother takes her daughter... or The mother leads her daughter...

Secum means with herself or more naturally here with her.

It is made from:

• se = herself / himself / themselves (a reflexive pronoun)
• cum = with

In Latin, when cum is used with me, te, se, nobis, vobis, it is attached to the end:

• mecum = with me
• tecum = with you
• secum = with himself / herself / themselves
• nobiscum = with us
• vobiscum = with you all

So filiam secum ducit means the mother leads her daughter with her.

Because with certain pronouns, Latin normally puts cum after the pronoun rather than before it.

So instead of:

• cum me
• cum te
• cum se

Latin usually says:

• mecum
• tecum
• secum

This is just the normal idiomatic Latin form.

It refers to the subject, so here it refers to mater.

That is because se is a reflexive pronoun. A reflexive pronoun points back to the subject of its own clause.

So:

• Mater filiam secum ad forum ducit = The mother leads her daughter with her
  = the daughter goes along with the mother

It does not mean that the daughter is somehow taking someone with the daughter.

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

So:

• ad forum = to the forum

Since forum is a second-declension neuter noun, its accusative singular is the same as its nominative singular:

• nominative: forum
• accusative: forum

That is why the form does not change visibly here, even though it is accusative.

Ad forum means to the forum or toward the forum.

The forum in Roman life was a public place in the city, often used for business, politics, law, and social activity.

So the phrase tells you the destination of the movement.

Classical Latin normally has no articles like English the and a/an.

So:

• mater can mean mother, a mother, or the mother
• filiam can mean a daughter or the daughter
• forum can mean a forum or the forum

You choose the most natural English wording from the context. In this sentence, the mother and her daughter sound most natural in English.

Because Latin uses case endings more than word order.

• mater is nominative = subject
• filiam is accusative = direct object

So the meaning is fixed by the endings:

• mater = the one doing the action
• filiam = the one receiving the action

In English, word order is very important. In Latin, endings usually do that job.

Yes, but it is also perfectly normal Latin.

The sentence is:

• Mater = subject
• filiam = object
• secum = with her
• ad forum = to the forum
• ducit = verb

Latin word order is more flexible than English because the endings show the grammar. A verb at the end, as in ducit, is very common in Latin.

So this sentence has a very natural Latin order. English, by contrast, usually needs a stricter order:
The mother leads her daughter with her to the forum.

Yes.

The Latin present tense can often be translated in more than one way in English:

• leads
• is leading
• sometimes does lead, depending on context

So Mater filiam secum ad forum ducit could be understood as either:

• The mother leads her daughter with her to the forum
• The mother is leading her daughter with her to the forum

Both are grammatically possible. The wider context would tell you which sounds best.

They come from different declensions:

• mater, matris = mother → third declension
• filia, filiae = daughter → first declension
• forum, fori = forum → second declension neuter

That is why their forms change in different ways:

• mater stays mater in the nominative singular
• filia becomes filiam in the accusative singular
• forum stays forum in the accusative singular because neuter second-declension nouns have the same nominative and accusative singular form