Brilleetuiet ligger på bordet.

Because Danish, like English, often makes compound nouns by joining words together.
Here:

• brille = glasses / spectacle
• etui = case
• brilleetui = glasses case
• brilleetuiet = the glasses case

So brilleetuiet is not unusual at all; it is just a compound noun with the definite ending added at the end.

In Danish, the definite article is very often added as an ending to the noun instead of being a separate word.

• et etui = a case

• etuiet = the case

• et bord = a table
• bordet = the table

So:

• brilleetuiet = the glasses case
• bordet = the table

This is one of the biggest differences from English.

That -et ending is the usual definite ending for many neuter nouns in Danish.

For example:

• et bord → bordet
• et etui → etuiet

Since brilleetui is built around etui, it is also a neuter noun:

• et brilleetui → brilleetuiet

So here -et means the, not past tense or anything like that.

Danish often prefers a position verb where English might simply use is.

• ligger = lies / is lying
• står = stands
• sidder = sits

With objects, Danish commonly chooses the verb based on the object’s position. A glasses case is thought of as lying on the table, so ligger sounds natural.

So:

• Brilleetuiet ligger på bordet = The glasses case is lying on the table / is on the table

You might hear er in some contexts, but ligger is the more natural choice here.

The sentence follows normal Danish main-clause word order:

• Brilleetuiet = subject
• ligger = verb
• på bordet = adverbial phrase showing location

So the structure is:

Subject + Verb + Place

That is very similar to normal English word order in a simple statement.

på is a preposition meaning on in this sentence.

• på = on
• bordet = the table

So på bordet means on the table.

This is straightforward and works much like English here.

You learn it from the article used with the noun:

• et brilleetui = a glasses case

If a noun takes et, it is neuter.
If it takes en, it is common gender.

Since the indefinite form is et brilleetui, the definite form becomes brilleetuiet.

Good question. In Danish, compounds often use a base form or stem that does not always match the most common standalone form exactly.

So although briller is the usual word for glasses as an item you wear, the compound is:

• brilleetui = glasses case

This kind of thing is very common in compounds and just has to be learned as part of the word.

A learner-friendly approximation is:

BRIL-leh-eh-TOO-ee-et

A few useful points:

• brille has two syllables
• etui is usually pronounced roughly like eh-TOO-ee
• the definite ending -et is added at the end

The whole word may sound long at first, but Danish speakers treat it as one normal noun.

Yes. That would mean A glasses case is lying on the table.

Compare:

• Brilleetuiet ligger på bordet = The glasses case is on the table
• Et brilleetui ligger på bordet = A glasses case is on the table

So the difference is simply definite versus indefinite.

Yes. Danish often puts another element first, and then the verb still stays in second position.

For example:

• På bordet ligger brilleetuiet.

This is still correct, but it sounds a bit more marked or literary, because the location is being emphasized first.

That is a good example of the Danish verb-second pattern in main clauses.