Avus diu in horto sedet et fabulam narrat.

Avus is nominative singular, the case used for the subject of the sentence.
The grandfather is the one doing the actions, so Latin uses avus.

Its dictionary form is also avus, so here the nominative looks the same as the basic vocabulary form.

Latin does not have articles like the or a/an.

So avus can mean grandfather, a grandfather, or the grandfather, depending on context.
Likewise, in horto can mean in a garden or in the garden.

The exact English article is usually supplied by context or by the translation you are given.

Horto is ablative singular.

After in, Latin uses:

• ablative for location: in the garden
• accusative for motion toward: into the garden

So:

• in horto = in the garden
• in hortum = into the garden

Because the sentence describes where the grandfather is sitting, Latin uses in horto.

Diu is an adverb, and it means for a long time or a long while.

It modifies sedet:
Avus diu sedet = The grandfather sits for a long time / has been sitting for a long time

It can also give a sense that the whole situation is prolonged, even though it stands right before in horto sedet.

Latin normally uses the simple present tense where English often prefers a progressive form.

So sedet can mean:

• he sits
• he is sitting

Latin usually does not need a separate form like is sitting.

The same applies to narrat:

• he tells
• he is telling

Both are possible depending on context.

Fabulam is accusative singular, the case used for the direct object.

The grandfather is telling a story, so story is the thing being told.
That makes it the direct object, so Latin uses the accusative:

• fabula = a story as a subject
• fabulam = a story as an object

So:

• fabula narrat would be wrong for he tells a story
• fabulam narrat is correct

Both are 3rd person singular present active indicative.

That means each one means he/she/it ...s or is ...ing.

• sedet = he sits / he is sitting
• narrat = he tells / he is telling

They are 3rd person singular because the subject avus is one person.

Latin verbs usually already show the person and number, so a subject pronoun is often unnecessary.

For example:

• sedet already means he/she/it sits
• narrat already means he/she/it tells

Because avus is stated explicitly, Latin does not need is (he).
If a pronoun were used, it would usually be for emphasis or contrast.

Yes. Et means and, and here it links two verbs with the same subject:

• sedet
• narrat

The subject avus applies to both verbs:

The grandfather sits ... and tells a story.

Latin often states the subject once and then gives multiple verbs connected by et.

No, Latin word order is much freer than English word order because the endings show the grammatical roles.

So this sentence could be rearranged in various ways without changing the basic meaning, for example:

• Avus diu in horto sedet et fabulam narrat.
• In horto avus diu sedet et fabulam narrat.
• Avus fabulam narrat et diu in horto sedet.

However, different word orders can change emphasis or style.

The given order is natural and clear:

• subject first: Avus
• adverb and place: diu in horto
• first verb: sedet
• second action: et fabulam narrat

That placement is very natural in Latin. Latin often puts related details before the verb, especially adverbs and prepositional phrases.

So diu in horto sedet presents the circumstances first:

• for a long time
• in the garden
• he sits

Latin often saves the verb for a later position, sometimes even the end of the clause.

Grammatically, it most directly modifies sedet, because it stands near that verb.

So the clearest reading is:

The grandfather sits in the garden for a long time and tells a story.

But in actual context, a listener may also feel that diu colors the overall scene, suggesting a long, leisurely period during which he is sitting and telling the story.

The immediate grammar points most strongly to sedet.

Here are the basic forms:

• avus — grandfather
• diu — for a long time
• in — in, into (depending on case)
• hortus → horto here — garden
• sedeo → sedet here — sit
• et — and
• fabula → fabulam here — story
• narro → narrat here — tell, relate

This is useful because in Latin the form you see in the sentence is often changed by case or verb ending.

Yes. That is very normal.

Latin present tense often covers both:

• he tells a story
• he is telling a story

English chooses between those based on style and context, but Latin usually just uses the simple present form narrat.

So both English translations can be correct.