Learn Hardcore Greek: Αν έχεις πυρετό, είναι καλό να μείνεις σπίτι. - If you have a fever, it is good to stay at home.

Αν means if.
It is a shortened, more colloquial form of εάν; in modern spoken Greek they are practically interchangeable here.
So Αν έχεις πυρετό and Εάν έχεις πυρετό mean the same thing, with εάν sounding a bit more formal or careful.

Why is έχεις πυρετό used for you have a fever instead of something like είσαι άρρωστος?

Greek usually uses έχω (to have) with illnesses or symptoms:

έχω πυρετό = I have a fever
έχω πονοκέφαλο = I have a headache

Είσαι άρρωστος means you are sick/ill more generally, not specifically that you have a fever. The sentence focuses on the fever symptom, so έχεις πυρετό is natural.

Why is it πυρετό and not πυρετός?

The base noun is ο πυρετός (fever) in the nominative case.
After έχω, the noun is used as a direct object in the accusative case:

nominative: ο πυρετός
accusative: τον πυρετό (with the article), or just πυρετό (without the article)

So in έχεις πυρετό, πυρετό is the accusative form, which is required after έχεις.

What tense and person is έχεις?

Έχεις is present tense, active voice, indicative mood, 2nd person singular of έχω (to have).
So it literally means you have.

Why is there a comma after πυρετό?

The sentence is a conditional:

Αν έχεις πυρετό = the if-clause (condition)
είναι καλό να μείνεις σπίτι = the main clause (result/advice)

When the if-clause comes first in Greek, it is usually followed by a comma, just like in English: If you have a fever, it’s good to stay home.
If you reverse the order, the comma is usually not needed: Είναι καλό να μείνεις σπίτι αν έχεις πυρετό.

What does the structure είναι καλό να... mean exactly?

Είναι καλό να... literally means it is good to... or it is good that....
It introduces a general recommendation or evaluation, just like English it’s good to stay home, it’s good to drink water, etc.
It is followed by να plus a verb in the subjunctive, here να μείνεις (to stay).

What form of the verb is μείνεις, and why isn’t it μένεις?

Μείνεις is the aorist subjunctive, 2nd person singular, of the verb μένω (to stay).
After να, Greek uses the subjunctive mood. You can choose between:

να μείνεις (aorist subjunctive) – focus on the action as a whole, often one-time or bounded: to stay (this time)
να μένεις (present subjunctive) – focus on ongoing or repeated action: to be staying / to keep staying (regularly)

In this context, να μείνεις sounds like advice for a specific situation: (in that case) it’s good for you to stay home.

Why do we need να before μείνεις?

Modern Greek no longer has a true infinitive (like English to stay).
Instead, it uses να + subjunctive verb to express things that English often expresses with to + verb or that-clauses.
So να μείνεις corresponds to to stay / (that you) stay, and είναι καλό να μείνεις = it is good to stay.

Why is there no article before σπίτι? Could we say στο σπίτι instead?

With the noun σπίτι (house / home), Greek often drops the article when talking about home in a general sense, especially with verbs of movement or staying:

μένω σπίτι = I stay (at) home
πάω σπίτι = I go home

You can also say στο σπίτι (at the house / in the house) and it is correct:

είναι καλό να μείνεις στο σπίτι

Σπίτι without an article feels a bit more like home, whereas στο σπίτι is slightly more concrete, like in the house / at the house, but in everyday usage they are very close in meaning here.

Can the word order change, for example: Είναι καλό να μείνεις σπίτι αν έχεις πυρετό?

Yes. Greek word order is fairly flexible.
Both are correct and natural:

Αν έχεις πυρετό, είναι καλό να μείνεις σπίτι.
Είναι καλό να μείνεις σπίτι αν έχεις πυρετό.

Changing the order doesn’t change the basic meaning; it only slightly changes emphasis (starting either with the condition or with the advice).

How would this sentence change if I were talking to more than one person or being polite?

You would use the 2nd person plural, which is also the polite form:

Αν έχετε πυρετό, είναι καλό να μείνετε σπίτι.

Here:

έχετε = you (plural/polite) have
μείνετε = you (plural/polite) stay (aorist subjunctive)

The rest of the sentence stays the same.

Can this sentence refer to the future, like If you get a fever (later), it’s good to stay home?

Yes. Greek often uses the present tense in αν-clauses to talk about future situations as well as present ones.
So Αν έχεις πυρετό, είναι καλό να μείνεις σπίτι can mean:

If you have a fever (now), it’s good to stay home, or
If you get a fever (at any time / in the future), it’s good to stay home.

Using θα inside the αν-clause (αν θα έχεις) is usually incorrect in standard Greek.