Et mater et pater domi adsunt.

This is a very common Latin way to say both ... and ...

So:

• et mater et pater = both mother and father
• literally, both mother and father are present/at home

Latin can also simply say mater et pater, just as English says mother and father. But repeating et gives a slightly more balanced, emphatic structure: both the mother and the father.

They are both in the nominative singular.

That is because they are the subjects of the verb adsunt. In other words, they are the people who are present / are at home.

So:

• mater = mother as subject
• pater = father as subject

Even though there are two singular nouns, together they make a compound subject, which is why the verb is plural.

Because the sentence has two subjects joined together:

• mater
• pater

Together they mean mother and father, so Latin treats them as a plural group.

That is why the verb is:

• adsunt = they are present

Compare:

• mater adest = mother is present
• pater adest = father is present
• mater et pater adsunt = mother and father are present

Adsunt is the 3rd person plural present tense of adsum.

adsum is made from:

• ad- = to, near, at
• sum = I am

So adsum literally has the sense of be near, and in normal English it often means:

• be present
• be here
• sometimes, depending on context, be at hand

In this sentence, because of domi, the meaning is naturally something like are at home or are present at home.

Its present-tense forms are:

• adsum = I am present
• ades = you are present
• adest = he/she/it is present
• adsumus = we are present
• adestis = you all are present
• adsunt = they are present

Both are possible in Latin, but they are not exactly the same.

• sunt = they are
• adsunt = they are present / they are here / they are nearby

So adsunt adds the idea of presence.

In this sentence, domi adsunt gives a natural sense of they are at home or they are home. If Latin used only sunt, the sentence would be more bare: simply they are.

Domi means at home.

This is a special form called the locative, which is used for place where with a few words, especially names of towns/cities and a small number of common nouns such as domus.

So:

• domi = at home
• domum = homeward, to home
• domo = from home

This is something English speakers often have to get used to, because Latin does not always use a preposition where English does.

So Latin says:

• domi = at home

not literally something with a separate word for at.

Historically and grammatically, it is a noun form: the locative of domus.

But when you read it in a sentence, it often behaves a lot like an adverbial expression, because it tells you where something happens:

• domi adsunt = they are at home

So the best way to think about it is:

• grammatically: a special case form of domus
• functionally: an expression of place, like at home

Yes. Latin word order is much more flexible than English word order because the endings tell you the grammatical roles.

So all of these are possible, with slightly different emphasis:

• Et mater et pater domi adsunt
• Mater et pater domi adsunt
• Domi mater et pater adsunt
• Adsunt domi mater et pater

The version you have is very natural. It places domi before the verb, which makes the location part of the statement flow smoothly: mother and father are at home.

In Latin, word order often affects emphasis more than basic grammatical meaning.

Latin does not have articles like English the and a/an.

So:

• mater can mean mother, a mother, or the mother
• pater can mean father, a father, or the father

The context tells you which English wording is best.

That is why a Latin sentence can often be translated into English in more than one natural way.

They are common nouns of the third declension, and learners often notice that they look a little unusual because the nominative forms end in -er.

Their important forms are:

• mater, matris = mother
• pater, patris = father

So the nominative singulars are:

• mater
• pater

But the stems are seen in the other cases:

• matr-
• patr-

For example:

• matris = of the mother
• patris = of the father

So they are not wildly irregular, but you do need to learn their principal forms because the nominative does not show the full stem clearly.

Yes.

Because of the repeated et ... et, the Latin especially suggests both ... and ..., so Both mother and father are at home is often the best translation.

But in normal English, Mother and father are at home may also be a perfectly good translation, depending on how strongly you want to bring out the emphasis.

So the repeated et gives a little extra force, but the basic meaning stays the same.