Kartoffelsalaten er koldt tilbehør.

Because -en is the usual definite ending for a singular common-gender noun in Danish.

• en kartoffelsalat = a potato salad
• kartoffelsalaten = the potato salad

In this sentence, the noun is definite, so Danish attaches the definite article to the end of the noun instead of using a separate word like English the.

Danish normally writes compound nouns as a single word.

So:

• kartoffel
  • salat → kartoffelsalat

This is very common in Danish, much more so than in English. English often writes these as two words, but Danish usually joins them into one compound noun.

Because koldt belongs with tilbehør, and tilbehør is a neuter noun: et tilbehør.

In Danish, adjectives often agree with the noun they describe:

• common gender: kold
• neuter: koldt

So:

• koldt tilbehør = cold side dish / cold accompaniment

The adjective is not directly describing kartoffelsalaten here. It is describing the noun tilbehør.

They are close, but not exactly the same.

• Kartoffelsalaten er kold = The potato salad is cold.
  This describes its temperature/state directly.

• Kartoffelsalaten er koldt tilbehør = The potato salad is a cold side dish.
  This describes what kind of thing it is.

So the second sentence classifies potato salad as a type of accompaniment, while the first simply says it is cold.

Because tilbehør is often used in a more general, category-like way in Danish, especially in food contexts.

So koldt tilbehør can mean something like:

• cold accompaniment
• cold side dish as a category

Using et is also possible in many contexts:

• Kartoffelsalaten er et koldt tilbehør

That sounds a bit more like a cold side dish, emphasizing one item in a group.
Without et, the phrase feels slightly more general or classificatory.

Here, tilbehør is the predicate noun after er.

The sentence structure is:

• Kartoffelsalaten = subject
• er = verb
• koldt tilbehør = subject complement / predicate

So the sentence is built like X is Y.

This is the same general pattern as:

• Han er lærer
• Det er en god idé
• Suppen er varm

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

In this sentence:

• Kartoffelsalaten = first element
• er = second element

That is the normal Danish word order for a simple statement.

If you moved another element to the front, the verb would still stay in second position:

• Til middag er kartoffelsalaten koldt tilbehør.

So this sentence is behaving exactly as Danish main clauses usually do.

It is singular definite.

You can compare:

• en kartoffelsalat = a potato salad
• kartoffelsalaten = the potato salad
• kartoffelsalater = potato salads
• kartoffelsalaterne = the potato salads

So the sentence is talking about one specific potato salad.

Yes, that is a very helpful way to understand it.

In this sentence, koldt tilbehør works together as one noun phrase:

• koldt = adjective
• tilbehør = noun

So instead of reading it as if koldt must match kartoffelsalaten, it is better to read it as:

• [koldt tilbehør]

That makes the grammar much clearer.

Yes. Danish very often uses er plus a noun phrase to classify something.

For example:

• Tomaten er en frugt.
• Ris er godt tilbehør.
• Silden er kold mad.

Your sentence follows that same pattern. It is a very normal way to say what category something belongs to.