Elven er grensen.

In this sentence, -en is the definite article attached to the noun.

So:

• elv = river
• elven = the river

and

• grense = border
• grensen = the border

Unlike English, Norwegian often puts the at the end of the noun instead of using a separate word.

Because the sentence is talking about a specific river and a specific border:

• Elven = the river
• grensen = the border

The sentence is identifying one thing as another: the river is the border. That makes both nouns definite in a very natural way.

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

So:

• å være = to be
• er = am / is / are depending on context

In Elven er grensen, er means is.

No, this is very straightforward. The word order is:

• Elven = subject
• er = verb
• grensen = subject complement / predicate noun

So it works much like English:

• The river is the border.

This is a basic subject + verb + complement sentence.

Because grensen means the border, while grense means just border or a border depending on context.

In this sentence, the river is being identified as the specific border, not just a border in a general sense.

Compare:

• Elven er grensen. = The river is the border.
• Elven er en grense. = The river is a border.

The second sentence is possible, but it means something more general and is not the same idea.

Yes, in Bokmål that is also possible.

Many feminine nouns in Bokmål can appear in two styles:

• elven / grensen
• elva / grensa

So these can both work:

• Elven er grensen.
• Elva er grensa.

They mean the same thing. The difference is mainly grammatical style and preference.

For a learner, it is useful to know:

• elv is often feminine
• grense is often feminine
• but Bokmål often allows -en definite forms too

They are commonly treated as feminine in Bokmål, but Bokmål is flexible.

That means you may see:

• ei elv / elva
• ei grense / grensa

But you may also see the more common-gender-looking forms:

• en elv / elven
• en grense / grensen

So even if a noun is feminine, Bokmål often allows masculine/common-style forms as well.

The dictionary forms are the indefinite singular forms:

• elv = river
• grense = border

You may also see the indefinite articles:

• en elv or ei elv
• en grense or ei grense

Then the definite singular forms are:

• elven or elva
• grensen or grensa

Because Norwegian usually expresses definiteness by adding an ending to the noun.

So instead of saying:

• the river

Norwegian often says:

• elven

where the idea of the is built into the noun.

This is one of the biggest structural differences from English.

A more general version would be:

• En elv er en grense.

That means A river is a border or A river can be a border.

But Elven er grensen is much more specific. It refers to one particular river and one particular border, and says they are the same thing.