Rektorn kommer in i klassrummet och säger hej till varje elev.

Because rektorn is the definite form of rektor.

• rektor = a principal / headteacher
• rektorn = the principal / the headteacher

In Swedish, the definite article is often attached to the end of the noun instead of being a separate word like the in English.

So:

• Rektorn kommer in ... = The principal comes in ...

Kommer is the present tense of komma, meaning to come.

So kommer can mean:

• comes
• is coming, depending on context

In this sentence, it is most naturally understood as comes in or is coming in.

Because komma in is a very common verb expression meaning to come in or to enter.

• komma = to come
• komma in = to come in / enter

If you said only kommer, the idea of entering the room would be missing. The word in adds that movement from outside to inside.

This is a very common thing for English speakers to notice.

• in is a particle that belongs with the verb: komma in = come in
• i klassrummet tells you where the movement goes: into the classroom

So the whole phrase means something like:

• comes in, into the classroom

That may feel a little repetitive from an English point of view, but it is normal Swedish.

Because klassrummet is the definite form of klassrum.

• ett klassrum = a classroom
• klassrummet = the classroom

The noun klassrum is an ett-word, so its definite singular ending is -et.

So:

• i klassrummet = in the classroom

Because the subject is still the same: rektorn.

The sentence has two coordinated parts:

• Rektorn kommer in i klassrummet
• och säger hej till varje elev

In English, we do the same thing:

• The principal comes into the classroom and says hello to every student

We do not need to repeat the principal in the second part, because it is understood.

In Swedish, the normal expression is säga hej till någon, which means to say hi/hello to someone.

• säga = to say
• hej = hi / hello
• till någon = to someone

So:

• säger hej till varje elev = says hello to each student

The preposition till is required in this expression.

Varje means each or every.

After varje, Swedish uses a singular noun:

• varje elev = each student / every student

This is similar to English, where we also say:

• every student
• not every students

So elev stays singular here.

Because after varje, Swedish uses the indefinite singular form of the noun.

• elev = student
• eleven = the student

Since varje already works as the determiner, you do not add the definite ending.

So:

• varje elev = correct
• varje eleven = incorrect

The sentence follows normal Swedish main-clause word order.

First clause:

• Rektorn = subject
• kommer = finite verb
• in i klassrummet = rest of the clause

Second coordinated clause:

• och säger hej till varje elev

Swedish main clauses usually follow the V2 rule, meaning the finite verb comes in the second position. In this sentence, that gives us:

• Rektorn kommer ...

Since the second clause keeps the same subject, the subject is simply omitted.

Yes. That is a very natural translation.

Even though the Swedish literally looks more like The principal comes in into the classroom, natural English often prefers:

• The principal enters the classroom
• The principal comes into the classroom
• The principal comes in to the classroom

All of those can fit, depending on style.

Here hej means hi or hello.

So:

• säger hej till varje elev = says hi to each student / says hello to every student

In Swedish, hej is a very common everyday greeting. It is neutral and works in most situations.

In this sentence, yes, elev is best understood as student or pupil.

A useful distinction is:

• elev is often used for someone in school
• student in Swedish often refers more specifically to a university student, or in some contexts to someone graduating from upper secondary school

So in a classroom sentence like this, elev is exactly the word you would expect.