العجوز هنا.

Because this is a nominal sentence in Arabic. In the present tense, Arabic normally does not use a separate verb for is/am/are.

So العجوز هنا literally looks like:

• العجوز = the old person / the elderly one
• هنا = here

But it means The old person is here.

If you wanted was, then Arabic would use a verb, for example كان العجوز هنا = The old person was here.

It can be understood as either, depending on context, but here it functions like a noun: the old person or the elderly one.

The word عجوز can also be used adjectivally:

• رجل عجوز = an old man
• امرأة عجوز = an old woman

But in العجوز هنا, the word stands on its own with الـ, so it means something like the old one / the elderly person.

From this sentence alone, you usually cannot tell for sure.

One important thing about عجوز is that it is often used with the same form for both masculine and feminine:

• رجل عجوز = an old man
• امرأة عجوز = an old woman

So العجوز هنا can mean:

• The old man is here
• The old woman is here
• The elderly person is here

Context tells you which one is meant.

Because not all Arabic words that can refer to women need a ـة ending.

عجوز is one of those words whose form often stays the same for masculine and feminine. So Arabic does not make it feminine by changing it to عجوزة in Modern Standard Arabic here.

That means:

• رجل عجوز
• امرأة عجوز

both use عجوز.

الـ is the definite article, equivalent to the in English.

So:

• عجوز = an old person / an elderly person
• العجوز = the old person / the elderly one

In this sentence, the noun is definite, which is very natural for the subject of a nominal sentence.

In normal Modern Standard Arabic, عجوز هنا is not the most natural way to say that.

With an indefinite noun like عجوزٌ, Arabic often prefers:

• هنا عجوزٌ = There is an old person here
• يوجد عجوزٌ هنا = There is an old person here

So the given sentence uses العجوز because it is definite: the old person is here.

هنا means here, and it functions as the predicate of the sentence.

So grammatically:

• العجوز = the subject (مبتدأ)
• هنا = the predicate (خبر)

That is a very common Arabic sentence pattern:

• الكتاب هنا = The book is here
• الولد هناك = The boy is there
• العجوز هنا = The old person is here

A careful pronunciation is:

al-ʿajūzu hunā

In normal pause, you will often hear:

al-ʿajūz hunā

A few pronunciation notes:

• ع is the letter ʿayn, a sound English does not have.
• جو in عجوز has a long ū sound: jū
• هنا ends with a long ā: hunā

Because ع is a moon letter, not a sun letter.

In Arabic, الـ is pronounced fully before moon letters:

• القمر = al-qamar
• العجوز = al-ʿajūz

But with sun letters, the l sound assimilates:

• الشمس = ash-shams

So here you say al-ʿajūz, not aʿ-ʿajūz.

In full case marking, it would normally be:

العجوزُ هنا

with ـُ on العجوز because it is the subject of the nominal sentence.

So the full parsing is:

• العجوزُ = subject, nominative
• هنا = predicate

In everyday reading and speech, those final case vowels are often not pronounced, especially in pause, so you will usually just hear العجوز هنا.

Yes, but it changes the feel of the sentence.

• العجوز هنا = neutral statement: The old person is here
• هنا العجوز = more like Here is the old person or The old person is here with extra focus on here

So the original order is the more neutral, textbook pattern.

A common plural is عجائز.

So:

• عجوز = an old person / old man / old woman
• عجائز = old people / elderly women, depending on context

If you wanted The old people are here, you might say:

العجائز هنا

Again, context helps determine exactly who is meant.