Learn Hardcore Greek: Ο πελάτης περιμένει ήρεμα τον λογαριασμό στο εστιατόριο. - The customer is calmly waiting for the bill at the restaurant.

In Greek, the subject of the sentence (the one doing the action) normally appears in the nominative case. So we need the nominative article Ο in front of πελάτης.

Roughly:

Ο πελάτης = the customer (as subject of the sentence)

What form of the verb is περιμένει, and why don’t we see anything like “is waiting” with a separate word for “is”?

περιμένει is the 3rd person singular, present tense of the verb περιμένω (to wait).

περιμένω = I wait / I am waiting
περιμένεις = you wait / you are waiting
περιμένει = he/she/it waits / is waiting

Greek does not make a grammatical difference between simple present and present continuous as English does. One Greek present form covers both:

Ο πελάτης περιμένει
= The customer waits
or The customer is waiting

There is no separate “is” verb here; περιμένει already includes the idea of “is waiting.”

Why is there no preposition like “for” before τον λογαριασμό? In English we say “wait for the bill”.

In Greek, περιμένω normally takes a direct object without a preposition.

περιμένω κάποιον / κάτι
= I wait for someone / something

So you say:

περιμένω τον λογαριασμό
literally: I-wait the bill
meaning: I am waiting *for the bill*

Adding για (περιμένω για τον λογαριασμό) is either wrong or sounds unnatural in standard Greek in this context. The preposition for is simply built into how περιμένω works.

What exactly is ήρεμα? Is it an adjective or something else, and why not ήρεμος?

ήρεμα is an adverb, meaning calmly.

ήρεμος = calm (adjective, masculine form)
e.g. Ένας ήρεμος πελάτης = a calm customer
ήρεμα = calmly (adverb)
e.g. Περιμένει ήρεμα = He/She waits calmly

Very often in Greek, the adverb is formed from the masculine adjective ending in -ος by changing it to -α:

ήρεμος → ήρεμα (calm → calmly)
γρήγορος → γρήγορα (fast → quickly)
σιωπηλός → σιωπηλά (silent → silently)

We use the adverb because it describes how he waits (the manner of the action), not what kind of person he is.

Where can ήρεμα go in the sentence? Is περιμένει ήρεμα the only option?

Greek word order is fairly flexible, especially with adverbs. All of these are possible and natural, with only slight differences in emphasis:

Ο πελάτης περιμένει ήρεμα τον λογαριασμό στο εστιατόριο.
Ο πελάτης ήρεμα περιμένει τον λογαριασμό στο εστιατόριο.
Ο πελάτης περιμένει τον λογαριασμό ήρεμα στο εστιατόριο.

Usually, placing ήρεμα right after the verb (as in the original sentence) is the most neutral and common choice:

περιμένει ήρεμα = he waits calmly

Why is it τον λογαριασμό and not ο λογαριασμός?

τον λογαριασμό is in the accusative case, which is used for the direct object (the thing directly receiving the action of the verb).

The noun:

ο λογαριασμός = the bill (masculine, nominative, used as subject)
τον λογαριασμό = the bill (masculine, accusative, used as object)

In the sentence:

Ο πελάτης = subject (nominative)
περιμένει = verb
τον λογαριασμό = direct object (accusative)

That’s why the article changes from ο to τον.

Is the article τον really necessary before λογαριασμό? Could you just say περιμένει λογαριασμό?

You could say περιμένει λογαριασμό, and it’s grammatically correct, but it changes the nuance slightly.

περιμένει τον λογαριασμό
= he is waiting for the bill (a specific, known bill – the one from this meal)
περιμένει λογαριασμό
= he is waiting for a bill / for (some) bill (more general or indefinite)

In normal conversation about being at a restaurant, speakers almost always say τον λογαριασμό, because it’s a specific bill you are expecting.

What is στο exactly? Why not just σε or το?

στο is a contraction of:

σε + το → στο

σε is a preposition that often corresponds to in / at / on / to, depending on context.
το is the neuter singular definite article in the accusative.

So:

σε το εστιατόριο → στο εστιατόριο
= at the restaurant / in the restaurant

In modern Greek, these contractions are standard and almost always used:

σε + το → στο
σε + τα → στα
σε + τον → στον
σε + την → στην
σε + τις → στις

Why is εστιατόριο with το and not ο or η? What gender and case is it?

εστιατόριο is a neuter noun. Its basic form is:

το εστιατόριο = the restaurant (neuter, singular)

In this sentence we have:

στο εστιατόριο = σε + το εστιατόριο
→ at/in the restaurant

The case is accusative. In modern Greek, after σε, the noun is generally in the accusative, even when talking about location (at/in) rather than movement (to).

So:

σε + το εστιατόριο (accusative) → στο εστιατόριο
not στο εστιατόριου or anything else.

Can we move στο εστιατόριο to the beginning or elsewhere? How does that affect the meaning?

Yes, Greek word order is quite flexible. These are all possible:

Στο εστιατόριο, ο πελάτης περιμένει ήρεμα τον λογαριασμό.
Ο πελάτης στο εστιατόριο περιμένει ήρεμα τον λογαριασμό.
Ο πελάτης περιμένει ήρεμα στο εστιατόριο τον λογαριασμό.

The basic meaning (“The customer is calmly waiting for the bill at the restaurant”) stays the same.

The differences are mostly about emphasis:

Starting with Στο εστιατόριο emphasizes the location: At the restaurant, the customer is waiting...
Keeping στο εστιατόριο at the end, as in the original, is very natural and neutral in storytelling: you say who, what they’re doing, to what, and finally where.

The original sentence:

Ο πελάτης περιμένει ήρεμα τον λογαριασμό στο εστιατόριο.

is a very typical, neutral word order in Greek.