În cutia aceasta sunt cărți, iar în cutia de acolo sunt pahare.

Because Romanian usually puts the definite article at the end of the noun.

• cutie = box
• cutia = the box

So în cutia aceasta literally looks like in the box this, but it simply means in this box.

Yes, you could also say această cutie.

Both mean this box:

• cutia aceasta
• această cutie

The difference is mainly structural:

• cutia aceasta = noun with definite article + demonstrative after it
• această cutie = demonstrative before the noun, and the noun stays in its basic form

The postposed version (cutia aceasta) is very common in everyday Romanian.

Because aceasta has to agree with cutia in gender and number.

cutie is a feminine singular noun, so the demonstrative must also be feminine singular:

• aceasta / această = feminine singular
• acesta / acest = masculine singular

So:

• cutia aceasta = this box
• but with a masculine noun, you would use forms like acest or acesta

de acolo literally means from there, but in this kind of phrase it works like over there or the one there.

So:

• în cutia de acolo = in the box over there

Romanian could also say:

• în cutia aceea = in that box

The version with de acolo is a little more explicit and paints a clearer physical contrast: this box here vs that box over there.

Romanian often uses this word order in sentences that say what exists in a place.

So:

• În cutia aceasta sunt cărți
• literally: In this box are books

This sounds more natural in Romanian than starting with cărți here. The sentence first sets the location, then says what is there.

Romanian usually does not use a dummy word like English there.

English says:

• There are books in this box

Romanian simply says:

• În cutia aceasta sunt cărți

So sunt can mean just are, but in context it often corresponds to English there are.

Because the real subject is still cărți or pahare, and both are plural.

• cărți = books
• pahare = glasses

Romanian verbs agree with the subject even when the subject comes after the verb. So:

• sunt cărți = there are books
• sunt pahare = there are glasses

If the noun were singular, the verb would be singular too.

Because this sentence is introducing unspecified things inside the boxes.

• cărți = books
• pahare = glasses

In this kind of existential sentence, Romanian usually uses the indefinite form:

• sunt cărți = there are books
• sunt pahare = there are glasses

If you said cărțile or paharele, that would refer to specific books or glasses already known from context.

iar links two clauses and often adds a slight contrast or parallel comparison.

In this sentence, it is something like:

• and
• while
• whereas

So the idea is:

• In this box there are books, and in that box over there there are glasses

It is not a strong contradiction like but, but it does set the two boxes against each other in a neat parallel way.

A few useful ones here are:

• ă in cărți: a short neutral vowel, similar to the a in sofa
• î / â in În: a Romanian vowel with no exact English equivalent
• ț in cărți: pronounced ts

So cărți does not sound like English carts. The ț is clearly ts.

Also:

• În is pronounced with î
• iar is pronounced roughly like yar
• pahare has a clear h sound, unlike many English words where h can weaken or disappear