Learn Hardcore Icelandic: Vefsíðan sem hún notar til að læra íslensku er mjög skemmtileg. - The website that she uses to learn Icelandic is very fun.

sem introduces the relative clause sem hún notar til að læra íslensku (that she uses to learn Icelandic).
Inside that clause, sem stands for the object of the verb notar (uses).
In fuller, more literal terms: The website which she uses (it) to learn Icelandic.

In Icelandic, you must use sem in this kind of clause; you can’t drop it like in English:

English: The website she uses… (OK)
Icelandic: Vefsíðan sem hún notar… (correct)
*Vefsíðan hún notar… (incorrect)

Why does vefsíðan end in -an instead of just vefsíða?

Vefsíða means “a website” (indefinite).
Vefsíðan means “the website” (definite).

Icelandic usually marks definiteness by adding a definite article suffix to the noun:

vefsíða – a website (indefinite)
vefsíðan – the website (definite)

For feminine nouns ending in -a, the definite -n attaches to the vowel:

bók → bókin (the book)
kennsla → kennslan (the teaching)
vefsíða → vefsíðan (the website)

So in this sentence, we are talking about a particular, known website, so the definite form vefsíðan is used.

Why is the word order “Vefsíðan … er mjög skemmtileg” and not something closer to English?

The main clause of the sentence is:

Vefsíðan … er mjög skemmtileg.

Basic Icelandic main-clause order is Subject – Verb – (other stuff):

Vefsíðan (subject)
er (verb “is”)
mjög skemmtileg (predicate: “very fun/entertaining”)

The relative clause sem hún notar til að læra íslensku is just inserted after the subject noun to describe it:

Vefsíðan [sem hún notar til að læra íslensku] er mjög skemmtileg.

So the underlying structure is:

[Subject] Vefsíðan …
[Verb] er
[Complement] mjög skemmtileg

You could, for emphasis, move things around in ways that sound marked, but the sentence as given is the normal, neutral word order.

Why is it íslensku and not íslenska after að læra?

The noun íslen(s)ka (íslenska) means “Icelandic (the language)”.

Its main forms (singular) are:

Nominative: íslenska
Accusative: íslensku
Dative: íslensku
Genitive: íslensku

The verb að læra (to learn) normally takes its object in the accusative case.
So:

að læra íslensku = to learn Icelandic (Icelandic is the thing being learned)

Because íslenska has the form íslensku in the accusative, that’s what you see here.
You don’t see the case change in English, but Icelandic marks it on the noun.

What does the phrase til að mean, and why do we need both words?

til að here means roughly “in order to”, often translated simply as “to” in English:

til að læra íslensku
→ (in order) to learn Icelandic

How it works:

til on its own usually means “to / towards / for” in a directional or purposive sense.
Adding að plus an infinitive verb expresses purpose:
- Ég nota þetta til að læra. – I use this to learn / in order to learn.

You usually:

use að + infinitive after certain verbs (e.g. ég vil að fara is wrong; it should be ég vil fara),
but til að + infinitive specifically expresses purpose: for doing X, in order to do X.

So hún notar vefsíðuna til að læra íslensku = she uses the website in order to learn Icelandic.

Why is it hún notar and not something like hún er að nota for “she is using”?

Icelandic doesn’t use a continuous “be + -ing” form the same way English does.

hún notar vefsíðuna can mean:
- she uses the website (general habit), or
- she is using the website (right now), depending on context.

The form hún er að nota vefsíðuna does exist and literally means she is (in the process of) using the website right now, but:

English uses the continuous form very often.
Icelandic uses simple present much more where English would say “is doing”.

So in a neutral sentence like this, hún notar is the natural choice for “she uses / she is using”.

What tense is notar, and how is the verb að nota conjugated?

Notar is the present tense, 3rd person singular of að nota (to use).

Present tense of að nota:

ég nota – I use
þú notar – you (sing.) use
hann / hún / það notar – he / she / it uses
við notum – we use
þið notið – you (pl.) use
þeir / þær / þau nota – they use

So in the sentence:

hún notar = she uses / she is using

Why is the adjective skemmtileg, and not skemmtilegt or some other form?

Adjectives in Icelandic agree in gender, number, and case with the noun they describe.

The noun here is vefsíðan.
vefsíða is a feminine noun, nominative singular (subject).
Therefore, the adjective must also be feminine, nominative, singular.

The adjective is skemmtilegur (fun, entertaining). Its main nominative forms:

Masculine: skemmtilegur
Feminine: skemmtileg
Neuter: skemmtilegt

Since vefsíða is feminine:

Vefsíðan … er mjög skemmtileg.

If the noun were neuter, you would indeed see skemmtilegt, for example:

Það er mjög skemmtilegt. – It is very fun.
Námskeiðið er mjög skemmtilegt. – The course is very fun. (n.)

What is the role of mjög in er mjög skemmtileg, and where does it go in the sentence?

Mjög means “very” and it is an adverb that modifies the adjective skemmtileg.

skemmtileg – fun
mjög skemmtileg – very fun

Position:

Adverbs like mjög normally come before the adjective they modify.
So er mjög skemmtileg is the natural order: “is very fun”.

You generally would not say *er skemmtileg mjög; that sounds wrong.

Can I leave out sem, like in English “the website she uses”?

No. In this kind of relative clause, Icelandic requires the relative pronoun sem.

Correct: Vefsíðan sem hún notar…
Incorrect: *Vefsíðan hún notar…

In English, you can sometimes omit that/which/who in relative clauses when it’s the object:

The website (that) she uses… (OK in English)

In Icelandic, you keep sem:

vefsíðan sem hún notar – the website (that) she uses

Why is there no pronoun like hana (“it/her”) inside the clause sem hún notar til að læra íslensku?

The pronoun is basically built into the relative pronoun sem.

In full:

The clause means: which she uses (it) to learn Icelandic.

Inside the relative clause:

hún is the subject of notar.
sem stands for the object of notar (the thing being used = the website).

In English, we don’t say “the website that she uses it” – the that already refers to the website.
Similarly in Icelandic, we do not add an extra object pronoun:

Correct: Vefsíðan sem hún notar…
Incorrect: *Vefsíðan sem hún notar hana…

Is vefsíða feminine, and how does it decline?

Yes, vefsíða (website) is a feminine noun.

It follows a common feminine pattern (like kona “woman”, bók has a different pattern). A simplified singular declension (without the definite ending) is:

Nominative: vefsíða (subject)
Accusative: vefsíðu
Dative: vefsíðu
Genitive: vefsíðu

With the definite article suffixed:

Nominative: vefsíðan – the website
Accusative: vefsíðuna – the website (object)
Dative: vefsíðunni – to/for the website
Genitive: vefsíðunnar – of the website

In our sentence, vefsíðan is nominative definite because it’s the subject:
Vefsíðan … er mjög skemmtileg.

Could I say íslenskuna instead of íslensku, and what would be the difference?

You might sometimes see íslenskuna, but it carries a slightly different nuance.

að læra íslensku
– to learn Icelandic (in general)
að læra íslenskuna
– to learn the Icelandic language (more explicitly definite/specific)

Grammatically:

íslenska (nom.), íslensku (acc.), íslenskuna (acc. definite: the Icelandic).

In everyday usage:

að læra íslensku is by far the most common way to say “to learn Icelandic” as a language.
íslenskuna can feel more explicit, like stressing this particular language, Icelandic, but it’s not needed here.

So the sentence as given, with íslensku, is the neutral and most natural phrasing.