Under bordet ligger en tjock matta som gör rummet tystare.

In Swedish, the definite article is usually added to the end of the noun.

• bord = table
• bordet = the table
• rum = room
• rummet = the room

So:

• under bordet = under the table
• rummet = the room

If the nouns were indefinite, you would say:

• ett bord = a table
• ett rum = a room

Both bord and rum are ett-words, so their definite singular form ends in -et.

Swedish often moves a place phrase to the front when the speaker wants to emphasize location or use a more natural flow.

So these are both correct:

• En tjock matta ligger under bordet.
• Under bordet ligger en tjock matta.

The version in your sentence puts focus first on where the rug is.

Because Swedish follows the V2 rule in main clauses: the finite verb usually comes in the second position.

Here, Under bordet takes the first position, so the verb ligger must come next:

• Under bordet
  • ligger
    • en tjock matta

Even though en tjock matta is the subject, it comes after the verb because the first position is already occupied by Under bordet.

Ligger is the present tense of ligga, which means lies / is lying.

Swedish often uses special position verbs where English just uses is:

• ligga = lie, be lying
• stå = stand, be standing
• sitta = sit, be sitting

A rug is thought of as something lying flat on the floor, so ligger is the natural choice.

So Under bordet ligger en tjock matta sounds more natural than using är.

Yes, you could.

• Det finns en tjock matta under bordet = There is a thick rug under the table
• Under bordet ligger en tjock matta = Under the table lies/is a thick rug

The difference is mostly about style and focus:

• det finns introduces existence: there is
• ligger gives a more concrete picture of the rug’s position

Both are correct, but ligger is more vivid and specific.

Because matta is an en-word, not an ett-word.

So:

• en matta = a rug
• en tjock matta = a thick rug

In the indefinite singular, adjectives take different forms depending on the noun:

• with an en-word: tjock
• with an ett-word: tjockt

Compare:

• en tjock matta
• ett tjockt täcke

So ett tjockt matta is incorrect because matta is not an ett-word.

Here som means that, which, or that/which introducing a relative clause.

So:

• en tjock matta som gör rummet tystare

means:

• a thick rug that makes the room quieter

In Swedish, som is very commonly used for relative clauses, and it does not change the way English who/which/that can.

It describes matta.

The structure is:

• en tjock matta
• som gör rummet tystare

So the relative clause tells you something about the rug: it is the rug that makes the room quieter.

It would not make much sense for bordet to be the thing making the room quieter, so both grammar and meaning point to matta.

This uses the pattern:

göra + object + adjective/comparative

So:

• gör = makes
• rummet = the room
• tystare = quieter

Together:

• gör rummet tystare = makes the room quieter

This is a very common pattern in Swedish:

• Det gör mig glad. = It makes me happy.
• Regnet gör vägen hal. = The rain makes the road slippery.

Because tystare is the comparative form of tyst.

• tyst = quiet / silent
• tystare = quieter

The sentence says the rug makes the room quieter, not completely silent. That is a more natural idea, since a rug reduces noise rather than making a room totally silent.

For this adjective, Swedish forms the comparative with -are:

• tyst → tystare

Because for short, common adjectives like tyst, Swedish normally uses the -are comparative form.

So:

• tystare = correct, natural
• mer tyst = usually unnatural here

Using mer is more common with longer adjectives or in certain stylistic situations, but with tyst, tystare is the normal choice.

That version is not a good idea, because som gör rummet tystare could sound like it is attached to bordet, not matta.

A clearer version is:

• En tjock matta som gör rummet tystare ligger under bordet.

or the original:

• Under bordet ligger en tjock matta som gör rummet tystare.

These make it clear that the relative clause belongs to matta.