La tabaccaia apre presto e saluta il bigliettaio della stazione.

It means the person, specifically a female tobacconist / woman who works in or runs a tobacco shop.

The shop itself is tabaccheria.

So:

• la tabaccaia = the female tobacconist
• la tabaccheria = the tobacco shop

This is a very common point of confusion for English speakers.

They are the normal definite articles: la = the (feminine singular), il = the (masculine singular).

So:

• la tabaccaia = the female tobacconist
• il bigliettaio = the ticket clerk

Italian uses definite articles very often, including when talking about specific people in a situation like this. In English, learners sometimes expect to drop the, but in Italian it is usually required.

Because Italian normally does not need subject pronouns when the subject is already clear.

Here, the subject is explicitly given as La tabaccaia, so adding lei would be unnecessary.

Italian does this all the time:

• Apre presto = [She/He/It] opens early
• Saluta il bigliettaio = [She/He] greets the ticket clerk

Since the noun La tabaccaia is already there, the sentence sounds complete without lei.

They are both in the present indicative, third person singular.

• apre = opens
• saluta = greets / says hello to

In a sentence like this, the present tense can describe:

• a habitual action: she opens early and greets him regularly
• or a simple present description of what happens

Italian uses the present tense very broadly, much like English does in statements such as She opens early.

It can mean either, depending on context.

Here, with apre, it means early:

• apre presto = opens early

In other contexts, presto can also mean soon:

• A presto! = See you soon!

So the exact meaning depends on the verb and the situation.

Because that is a very natural position for an adverb in Italian.

• apre presto sounds normal and neutral

Italian word order is often flexible, but verb + adverb is very common. English speakers may want to analyze this too rigidly, but here it is simply the most natural phrasing.

Because salutare takes a direct object, not an indirect one.

So:

• salutare qualcuno = to greet someone

That is why you say:

• saluta il bigliettaio

not:

• saluta al bigliettaio

English can also use a direct object here: She greets the ticket clerk.

If the verb required a, then you would use al / alla / ai, etc., but salutare does not.

della is a contraction of di + la.

So:

• della stazione = literally of the station

In smoother English, that often becomes:

• the station ticket clerk
• or the ticket clerk at the station

This phrase specifies which ticket clerk we mean:

• il bigliettaio della stazione = the ticket clerk of/at the station

Italian often uses di + noun where English might prefer a noun-noun combination or at/from/of.

Yes. Bigliettaio is related to biglietto (ticket).

A bigliettaio is a male ticket clerk / ticket seller. The feminine form is bigliettaia.

So:

• biglietto = ticket
• bigliettaio = male person who sells/checks tickets
• bigliettaia = female person who sells/checks tickets

That word-building pattern is very common in Italian.

These are profession nouns with masculine and feminine forms:

• tabaccaio = male tobacconist

• tabaccaia = female tobacconist

• bigliettaio = male ticket clerk
• bigliettaia = female ticket clerk

So in this sentence:

• la tabaccaia is feminine
• il bigliettaio is masculine

This is useful because Italian often marks gender directly in the noun, article, and sometimes adjective.

A rough guide is:

• tabaccaia ≈ ta-bak-KA-ya
• bigliettaio ≈ bee-lyet-TA-yo

A few pronunciation points:

• In tabaccaia, the cc before a is a hard k sound.
• In bigliettaio, gli is a special Italian sound, roughly similar to the lli in million, though not exactly the same.
• The ending -aio / -aia is common in Italian profession words.

If you want, you can think of them approximately as:

• tabaccaia: four beats
• bigliettaio: four beats

That is not a perfect phonetic description, but it is a helpful start for English speakers.