Stigavörðurinn skrifar niður stigafjöldann.

Skrifar is the 3rd person singular present tense of skrifa (to write).

It matches the subject Stigavörðurinn, which is singular: The scorekeeper writes.
So the basic pattern is:

• skrifa = to write
• ég skrifa = I write
• hann / hún / það skrifar = he / she / it writes

In this sentence, skrifar means writes.

The ending -inn is the definite article in Icelandic, equivalent to English the.

So:

• stigavörður = a scorekeeper
• stigavörðurinn = the scorekeeper

In Icelandic, the is usually attached to the end of the noun instead of being a separate word.

Also, Stigavörðurinn is in the nominative singular, because it is the subject of the sentence.

Because it is the direct object of the verb, and after skrifa niður here, the object is in the accusative case.

The base forms are:

• stigafjöldi = number of points / point total
• stigafjöldinn = the point total (nominative)
• stigafjöldann = the point total (accusative)

So the sentence uses stigafjöldann because the scorekeeper is writing down that point total.

Here niður is part of the expression skrifa niður, which means to write down or to note down.

So this is similar to English phrasal verbs:

• write = skrifa
• write down = skrifa niður

In this sentence, niður is not mainly about physical downward movement. It is part of the meaning of the whole verb phrase.

Here it means point or score point, not step.

The noun stig can mean several things depending on context, including:

• step
• level
• grade
• point

In sports or scoring contexts, stig very often means points. That is the sense used in both stigavörðurinn and stigafjöldann.

In compounds, Icelandic often uses a combining form of a noun rather than the plain dictionary form. Here that form is stiga-.

So:

• stig = point
• stiga- = point-/of points, used in compounds

This is very common in Icelandic. Historically, this often comes from a genitive form, and for learners it is usually best just to recognize stiga- as the normal compound form here.

Yes, both are compound nouns.

You can break them down like this:

• stigavörðurinn = stiga-

  • vörður
    • -inn
      → point/score-keeper-the

• stigafjöldann = stiga-

  • fjöldi (declined here as -fjöldann)
    → point-total / number of points, in the definite accusative form

A very useful Icelandic reading strategy is to understand compounds from right to left:

• vörður tells you the word is a kind of keeper/guard
• fjöldi tells you the word is a kind of number/amount

The part before that narrows the meaning.

The sentence has a very normal Icelandic main-clause order:

• Stigavörðurinn = subject
• skrifar = finite verb
• niður = particle
• stigafjöldann = object

So the pattern is:

Subject + Verb + Particle + Object

This also fits Icelandic’s general verb-second tendency in main clauses. In a simple statement like this, that gives a very natural order.

Sometimes particles can move, but skrifa niður X is a very common and natural pattern, and it is the safest one to learn first.

So:

• skrifa niður stigafjöldann = very natural

You may also come across other placements in real Icelandic, but the version in your sentence is completely standard and learner-friendly.

A few useful pronunciation points:

• Stress usually falls on the first syllable of each word.
• j is pronounced like English y.
• ð is usually like th in this, or even softer.
• ö is a rounded vowel that English does not really have; it is similar to German ö.

So some tricky pieces are:

• niður: the ð is soft, like th in this
• vörðurinn: both the ö and ð may feel unfamiliar
• stigafjöldann: the jö sound may take practice

A rough English-style guide could be:

STI-ga-vur-thur-inn SKRIF-ar NI-thur STI-ga-fyol-dan

That is only approximate, but it can help you get started.