Puella matrem redeuntem salutat.

Because of their case endings:

• puella is nominative singular, the case normally used for the subject
• matrem is accusative singular, the case normally used for the direct object

So puella is the one doing the greeting, and matrem is the one being greeted.

Mater is the dictionary form and the nominative singular form.

But in this sentence, mother is the direct object of salutat (greets), so Latin puts it in the accusative:

• nominative: mater
• accusative: matrem

So matrem means mother in the form required by the sentence.

Salutat is a verb meaning greets.

More specifically, it is:

• present tense
• active voice
• indicative mood
• 3rd person singular

So it means he/she/it greets.
Since the subject is puella (girl), the full sense is the girl greets.

Redeuntem is the present active participle of redeo, redire (to return, to come back).

A participle is a verbal adjective. That means it has:

• a verb-like meaning: returning
• an adjective-like function: it describes a noun

Here, redeuntem describes matrem, so it means the mother who is returning or the returning mother.

Because it agrees with matrem.

Since redeuntem describes matrem, it must match it in:

• gender: feminine
• number: singular
• case: accusative

So:

• matrem = feminine singular accusative
• redeuntem = feminine singular accusative

This is a very common Latin pattern: adjectives and participles agree with the nouns they modify.

It means returning.

That is because it is a present participle, and the Latin present participle usually shows action happening at the same time as the main verb.

So the idea is:

• the girl greets the mother
• while the mother is returning

If Latin wanted having returned, it would normally use a different construction.

Yes. That is a very natural translation.

Other good translations are:

• The girl greets her mother as she returns.
• The girl greets her mother, who is returning.
• The girl greets the returning mother.

The exact English wording depends on style, but the Latin structure is the same.

Not by itself.

Matrem just means mother in the accusative singular. Latin does not explicitly say her here.

However, in context, English will often naturally say her mother, because:

• family relationships often imply possession
• Latin often leaves such possession unstated when it is obvious

So the girl greets her mother is usually a very reasonable translation, but the Latin word itself does not contain her.

It can, but Latin very often prefers a participle where English might use a relative clause.

So instead of saying:

• matrem quae redit = the mother who is returning

Latin can say more compactly:

• matrem redeuntem = the returning mother / the mother who is returning

This participial style is very common in Latin.

Yes. Latin word order is much more flexible than English word order because the endings show each word's role.

So all of these can mean basically the same thing:

• Puella matrem redeuntem salutat.
• Matrem redeuntem puella salutat.
• Puella salutat matrem redeuntem.

The differences are mostly about emphasis or style, not basic grammar.

Because Latin often places a modifying adjective or participle after the noun, though it can also come before it.

So:

• matrem redeuntem = the returning mother
• redeuntem matrem would also be possible

The position does not change the basic grammar, because the agreement in case, number, and gender makes the relationship clear.

They are:

• puella, puellae = girl
• mater, matris = mother
• redeo, redire = return, go back
• saluto, salutare = greet

And the forms in the sentence are:

• puella = nominative singular
• matrem = accusative singular
• redeuntem = accusative feminine singular present participle
• salutat = 3rd person singular present active indicative