Terroristen er i byen.

Because Norwegian often adds the definite article to the end of the noun.

• terrorist = terrorist / a terrorist
• terroristen = the terrorist

So terroristen er i byen literally means the terrorist is in the city/town.

There is no separate word for the here, because Norwegian usually puts definiteness on the noun itself as a suffix.

• terroristen = the terrorist
• byen = the city / the town

This is a very common feature of Norwegian nouns.

For the same reason: byen is the definite form.

• by = city / town or a city / a town
• byen = the city / the town

In this sentence, i byen is a very common expression that often means in town, not necessarily inside the city in a very literal sense.

This is just a difference between the languages. Norwegian commonly uses the definite form in this expression:

• i byen = in town

Even though byen literally looks like the town/city, the whole phrase is often translated more naturally as in town.

Er means is / am / are, depending on the subject.

It is the present tense of the verb å være (to be).

Examples:

• jeg er = I am
• du er = you are
• terroristen er = the terrorist is

Unlike English, Norwegian does not change the present-tense form of this verb for different persons in the same way. Er stays the same.

This is the normal basic Norwegian sentence pattern:

subject + verb + other information

So:

• Terroristen = subject
• er = verb
• i byen = place expression

This is very similar to normal English word order:

• The terrorist is in town

Yes, that is possible, but the meaning and emphasis change slightly.

• Terroristen er i byen = neutral statement
• I byen er terroristen = more emphasis on in town

Norwegian often allows a different element to come first for emphasis, but then the verb still comes in second position. This is part of the V2 word order rule.

It is singular.

• terroristen = the terrorist (singular)

The plural would be different:

• terrorister = terrorists
• terroristene = the terrorists

So Terroristen er i byen refers to one terrorist.

You would say:

En terrorist er i byen.

Here:

• en terrorist = a terrorist
• terroristen = the terrorist

So the sentence changes depending on whether the noun is indefinite or definite.

Because the sentence is definite, not indefinite.

Compare:

• En terrorist er i byen = A terrorist is in town
• Terroristen er i byen = The terrorist is in town

In Norwegian, if a noun is indefinite singular, it often uses en, ei, or et depending on gender. But if it is definite, the noun usually gets a suffix instead.

Yes. By can mean city or town, depending on context.

So:

• i byen can be understood as in the city
• but very often it is best translated idiomatically as in town

The exact English translation depends on the situation.

A rough English-friendly guide would be:

teh-ro-RIS-ten ehr ee BY-en

A few notes:

• r is usually pronounced more clearly than in many English accents
• y in byen is not exactly like English y; it is a front rounded vowel, which many English speakers find tricky
• byen is often pronounced something like BY-en, but the exact sound varies a bit by dialect

If you want to sound more natural, it helps to learn the vowel in by separately, since it is a very common Norwegian word part.