Tannlæknirinn segir að ég þurfi að koma aftur eftir viku.

In Icelandic, the definite article is usually attached to the end of the noun.

• tannlæknir = dentist
• tannlæknirinn = the dentist

So the ending -inn is the equivalent of English the here.

Also, tannlæknir is a compound word:

• tann- = tooth / dental
• læknir = doctor

So it is literally something like tooth-doctor.

They are two different words that just happen to be spelled the same.

• The first að means that and introduces a subordinate clause:
  segir að ég þurfi... = says that I need...
• The second að is the infinitive marker to:
  að koma = to come

So they do different jobs:

• að = that
• að = to

Because once you enter a subordinate clause introduced by að, Icelandic normally stops using main-clause verb-second word order.

In a main clause, Icelandic often puts the finite verb early:

• Ég þarf... = I need...

But in a subordinate clause, the subject usually comes before the finite verb:

• að ég þurfi...

So að ég þurfi is the normal structure here.

Þurfi is the present subjunctive form of þurfa (to need / have to).

This is common in subordinate clauses after verbs like segja when the sentence is reporting what someone says, thinks, claims, or recommends rather than presenting it as a plain direct fact from the speaker.

So:

• ég þarf = I need (indicative)
• að ég þurfi = that I need / that I should need in reported-speech style (subjunctive)

For a learner, the important point is: segir að + subjunctive is a very common pattern.

Because þurfa is followed by að + infinitive.

So the pattern is:

• þurfa að gera eitthvað = need to do something

In this sentence:

• þurfi að koma = need to come

This is actually closer to English need to than to English must.

Koma is the infinitive: to come.

The only finite verb in that part of the clause is þurfi. After it, Icelandic uses the infinitive phrase:

• þurfi að koma = need to come

So you do not conjugate koma here. It stays in the dictionary form.

It can mean both again and back, depending on context.

With koma (come), koma aftur very often means:

• come back
• or more literally, come again

In this sentence, English usually prefers come back, but again is also part of the idea.

Eftir means after, and viku is the form of vika (week) used after this preposition.

So:

• eftir viku = after a week
• in natural English, often in a week or a week later, depending on context

A useful thing to remember:

• eftir viku = after a week / one week later
• í viku = for a week

So eftir viku is about when, not duration.

Because Icelandic very often uses present-tense forms to talk about the future when the context already makes the time clear.

Here, the future meaning comes from the whole idea:

• the dentist says
• I need to come back
• after a week

So Icelandic does not need a separate future form to express will need to come back. The context already does that.

Because this Icelandic sentence is built as The dentist says that I need..., not The dentist tells me to...

So the structure is:

• Tannlæknirinn segir = The dentist says
• að ég þurfi... = that I need...

Here, ég is the subject of the subordinate clause.

If you wanted a structure closer to The dentist tells me to come back in a week, Icelandic could say something like:

• Tannlæknirinn segir mér að koma aftur eftir viku.

That is a different structure:

• segir mér = tells me / says to me
• að koma = to come

So the original sentence focuses on what the dentist says about your need, not directly on giving you an instruction.