Sagen er vigtig.

Because -en is the definite singular ending for many common-gender Danish nouns.

• en sag = a case / a matter / an issue
• sagen = the case / the matter / the issue

So Sagen er vigtig literally means The matter is important.

Unlike English, Danish usually adds the definite article to the end of the noun instead of putting a separate word like the in front.

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

So:

• jeg er = I am
• du er = you are
• han/hun er = he/she is
• sagen er = the matter is

In this sentence, er simply means is.

Because adjective endings in Danish depend on the noun phrase and grammar around them.

Here, vigtig is the base/common form, and that is what you usually use in a sentence like this after er when the subject is a singular common-gender noun:

• En sag er vigtig.
• Sagen er vigtig.

Compare:

• Et problem er vigtigt.
  Here vigtigt is used because problem is a neuter noun (et problem).

• Sagerne er vigtige.
  Here vigtige is used because the subject is plural.

So in Sagen er vigtig, vigtig matches a singular common-gender noun.

No. Sag is a very common word with several related meanings, depending on context.

It can mean things like:

• case
• matter
• issue
• cause
• sometimes even thing in certain expressions

So Sagen er vigtig could be understood as:

• The matter is important
• The issue is important
• The case is important

The exact English word depends on context.

The word order is very straightforward:

• Sagen = subject
• er = verb
• vigtig = adjective complement

So the pattern is:

Subject + verb + complement

This is the normal order for a simple Danish statement, just like English:

• The matter is important.

That would be:

En sag er vigtig.

Compare:

• En sag = a matter
• Sagen = the matter

So the only difference is whether the noun is indefinite or definite.

In standard Danish, the g in sagen is usually not pronounced like a hard English g.

A learner will often hear something closer to:

• sa-yen
• or a very soft glide sound between the vowels

So although it is spelled sagen, the pronunciation is softer than many English speakers expect.

Also:

• a in sag is a long vowel
• -en is the attached definite ending

Exact pronunciation varies a bit by speaker and accent, but the important thing is that the g is typically soft.

Yes, but then the structure changes.

If the adjective comes before the noun, you would normally say:

• en vigtig sag = an important matter
• den vigtige sag = the important matter

But in your sentence, the adjective comes after the verb:

• Sagen er vigtig. = The matter is important.

So both are possible, but they are different sentence structures:

• adjective + noun: en vigtig sag
• noun + er + adjective: Sagen er vigtig

Because Danish usually expresses the by adding an ending to the noun.

So instead of:

• the matter

Danish often uses:

• sagen

This is called the postposed definite article.

Very roughly:

• en sag = a matter
• sag-en = matter-the = the matter

English speakers often need time to get used to this, because English puts the before the noun, while Danish often attaches it at the end.

Yes, Det er vigtigt means It is important or That is important.

The difference is:

• Sagen er vigtig = The matter is important
  This specifically says that the matter/case/issue is important.

• Det er vigtigt = It is important
  This is more general and does not name the thing directly.

So Sagen er vigtig is more specific.