Der ligger et tyndt hæfte i hendes taske, og hæftemaskinen ligger på skrivebordet derhjemme.

Literally, it means something like There lies a thin booklet in her bag.

In natural English, you would usually say:

There is a thin booklet in her bag.

Danish often uses location verbs such as ligge instead of the more general English be.

Because Danish very often prefers a position/location verb when talking about where something is.

Here, ligger comes from ligge, which means to lie or to be lying.

So instead of saying only it is in her bag, Danish often says the equivalent of:

• it lies in her bag
• it is lying in her bag

This is very common with objects that are understood as resting somewhere.

Compare some common Danish location verbs:

• ligge = lie, be lying
• stå = stand, be standing
• sidde = sit, be sitting

So a booklet in a bag and a stapler on a desk are both naturally described with ligger.

Here, Der is an expletive/dummy subject, like there in English there is.

So:

• Der ligger et tyndt hæfte ... = There is a thin booklet ...

This der does not mean there in the sense of a place. It is just part of the structure used to introduce that something exists or is located somewhere.

No, not really.

• The first Der is the dummy there in there is / there lies.
• In derhjemme, the whole word means at home.

So in this sentence:

• Der ligger ... = There is / there lies ...
• derhjemme = at home

Even though both contain der, they are doing different jobs.

Because hæfte is a neuter noun, and in Danish an adjective in the indefinite singular neuter usually takes -t.

So:

• en stor bil = a big car
• et stort hus = a big house

Here:

• et hæfte = a booklet
• et tyndt hæfte = a thin booklet

So tynd becomes tyndt because hæfte is an et-word.

In this sentence, hæfte means something like:

• booklet
• pamphlet
• exercise book
• thin booklet

The exact English word depends on context, but the important idea is that it is a small thin bound paper item.

Do not confuse it with hæftemaskine, which means stapler.

Yes, they are related in form.

• hæfte can refer to a booklet, and the word is historically connected with the idea of something fastened together.
• hæftemaskine literally means something like fastening machine, but in normal English it is simply stapler.

So in this sentence:

• et tyndt hæfte = a thin booklet
• hæftemaskinen = the stapler

They look similar, but they refer to different things.

Because Danish usually puts the definite article at the end of the noun.

So:

• en hæftemaskine = a stapler
• hæftemaskinen = the stapler

The noun hæftemaskine is a common-gender noun (an en-word), so its singular definite ending is -en.

For the same reason: Danish usually marks definiteness by adding an ending to the noun.

• et skrivebord = a desk
• skrivebordet = the desk

Since skrivebord is a neuter noun (an et-word), the singular definite ending is -et.

So:

• på skrivebordet = on the desk

Because hendes means her, referring to a female person, while sin is a reflexive possessive that normally refers back to the subject of the clause.

In this sentence, hendes is the natural choice because the bag belongs to her, and there is no suitable subject that sin could refer back to.

So:

• hendes taske = her bag

For English speakers, it is often safest to remember:

• hans / hendes / deres = his / her / their
• sin / sit / sine = his own / her own / its own / their own, when referring back to the clause subject

Because both objects are understood as resting somewhere.

In Danish, it is very normal to say that many inanimate objects lie somewhere:

• a booklet lies in a bag
• a stapler lies on a desk

English often just says is, but Danish is more likely to choose a specific position verb.

Derhjemme means at home.

In many contexts, hjemme and derhjemme are very similar. But derhjemme can sound a little more like:

• at home
• back home
• at their/her/his home

In this sentence:

• på skrivebordet derhjemme = on the desk at home

It helps locate the desk as being at home rather than somewhere else.

It is a normal main clause with standard Danish word order:

• hæftemaskinen = subject
• ligger = verb
• på skrivebordet derhjemme = place information

So literally:

• and the stapler lies on the desk at home

This is straightforward subject + verb + adverbial word order.

Yes, but it would change the structure and emphasis.

You could say:

• Et tyndt hæfte ligger i hendes taske.

That means:

• A thin booklet is lying in her bag.

This is perfectly grammatical, but it sounds less like an existential there is statement and more like you are talking specifically about the booklet.

So:

• Der ligger et tyndt hæfte ... = introduces the existence/location of something
• Et tyndt hæfte ligger ... = starts directly with the noun phrase

A rough guide:

• der ≈ dehr
• ligger ≈ LEE-gər
• tyndt ≈ toont with a short rounded vowel; this one is hard for English speakers
• hæfte ≈ HEF-də
• hendes ≈ HEN-əs
• taske ≈ TA-sgə
• hæftemaskinen ≈ HEF-tə-ma-skee-nən
• skrivebordet ≈ SKREE-və-bor-dəth
• derhjemme ≈ dehr-yem-ə

A few important points:

• æ is a very common Danish vowel and is not exactly like any single English vowel.
• Final -e in Danish is often a weak schwa sound, like uh.
• Danish pronunciation is often much less phonetic than spelling suggests, so listening to native audio is especially helpful.