Learn Hardcore Icelandic: Hún skrifar markmið sín í glósubókina til að muna þau betur. - She writes her goals in the notebook to remember them better.

Sín is the reflexive possessive pronoun: it means her own / his own / their own and always refers back to the subject of the clause.
Here the subject is Hún (she), so markmið sín means her own goals.

If you said markmið hennar, that would usually mean another woman’s goals (someone else’s, not the subject’s).

So:

Hún skrifar markmið sín … = She writes her own goals.
Hún skrifar markmið hennar … = She writes that other woman’s goals.

Why is it sín and not sína / sínar / sínum?

The form of sinn / sín / sitt has to agree with the noun in gender, number, and case.

Markmið is neuter plural accusative (because it is the direct object).
The reflexive possessive for neuter plural accusative is sín.

Some forms for comparison:

sinn – masc. sg. acc.
sína – fem. sg. acc.
sitt – neut. sg. acc.
sínar – fem. pl. nom./acc.
sínum – dat. pl. (all genders)

Here we need: neuter + plural + accusative → sín.

How do we know markmið is plural when it doesn’t change form?

The noun markmið is neuter and has the same form in nominative and accusative, both singular and plural.
We can see it is plural here because the pronoun referring to it later is þau, which is neuter plural.

það = neuter singular (it)
þau = neuter plural (they / them)

So markmið … þau tells us markmið is being understood as goals (plural), not just a goal.

Why is the pronoun þau used, and what does it refer to?

Þau is the neuter plural third-person pronoun, meaning they / them for neuter nouns.
It refers back to markmið (goals), which is a neuter noun understood in the plural here.

So til að muna þau betur literally means in order to remember them better, where them = her goals.

Why is it í glósubókina and not just í glósubók?

The ending -ina is the definite article attached to a feminine noun in the accusative singular:

glósubók = a notebook
glósubókin = the notebook (nominative)
glósubókina = the notebook (accusative)

Because í here expresses motion into something, it takes the accusative case, and we are talking about a specific notebook, so we use the definite, accusative form: glósubókina = into the notebook.

Why does í take the accusative here instead of the dative?

The preposition í can take either accusative or dative:

Accusative = motion into, change of location (into something)
Dative = location, being inside something (in something)

In this sentence she is putting (writing) her goals into the notebook (a movement into it), so í governs the accusative: í glósubókina.

If you were describing a static location, you’d use dative, e.g. Markmiðin eru í glósubókinni (The goals are in the notebook).

What is the role of til að in til að muna þau betur?

Til að + infinitive is a very common way of expressing purpose, similar to English “in order to” / “to”.

til = to, for
að introduces the infinitive verb (muna)

So til að muna þau betur means in order to remember them better or simply to remember them better.

Why is muna in the infinitive form?

After til að you use the infinitive form of the verb, not a conjugated form.
So:

hún man = she remembers (conjugated, present tense)
til að muna = in order to remember (infinitive)

Because this is a purpose clause, introduced by til að, we must use the infinitive muna.

Does muna always take an object like þau?

Muna typically takes an object in the accusative case:

Ég man það = I remember it.
Hún man þau = She remembers them.

In this sentence, þau (neuter plural accusative) is the object of muna.
You can sometimes see muna eftir used as well (with dative), but that is a slightly different construction and often a bit more like “remember something / someone (look back to)”. Here the straightforward muna + accusative is used.

Why is it Hún skrifar and not Hún skrifar að or something similar?

The verb skrifa is transitive and takes a direct object without any extra particle:

Hún skrifar markmið sín = She writes her goals.

There is no need for a preposition or conjunction with skrifa in this basic object construction.
The til að phrase that follows (til að muna þau betur) is not part of skrifa; it’s a separate purpose clause explaining why she writes them.

Could you change the word order of til að muna þau betur? For example, til að muna betur þau?

The natural word order is til að muna þau betur:

verb (muna)
object (þau)
adverb (betur)

You could sometimes move betur for emphasis in spoken language, but til að muna betur þau sounds unnatural in standard Icelandic.
Generally, keep verb – object – adverb in this type of infinitive purpose phrase.

Can the subject Hún be left out, like in some other languages?

No. Icelandic is not a “pro‑drop” language like Spanish or Italian.
You normally must state the subject pronoun unless it is very clearly understood from context in a special construction.

So you need Hún skrifar …, not just Skrifar markmið sín … in a neutral sentence.

Is glósubók a compound word, and what does it literally mean?

Yes, glósubók is a compound:

glósur = notes (as in class notes, study notes)
bók = book

So glósubók is literally a “notes-book”, which corresponds to notebook in English.

The form glósubókina in the sentence is definite, accusative singular: the notebook.