Amma vill prjóna nýja peysu fyrir dóttur mína í vetur.

Because vilja means to want / to want to, and it is followed by an infinitive.

• vill = wants
• prjóna = to knit

So Amma vill prjóna means Grandma wants to knit.

If you said Amma prjónar, that would mean Grandma is knitting / Grandma knits, which is a statement about what she does, not what she wants to do.

After modal-type verbs like vilja (want), Icelandic normally uses the infinitive without að.

So:

• vill prjóna = wants to knit
• not vill að prjóna

This is similar to English in a way, except English uses to, while Icelandic often leaves it out after certain verbs.

Vill is the 3rd person singular present form of vilja (to want).

It matches Amma, which is a third-person singular subject.

A few forms of vilja are:

• ég vil = I want
• þú vilt = you want
• hann/hún vill = he/she wants

So Amma vill is the correct match for Grandma wants.

Because peysu is in the accusative singular, and the adjective has to match it.

Here peysu is the direct object of prjóna: it is the thing being knitted.

• nominative: ný peysa = a new sweater (as subject)
• accusative: nýja peysu = a new sweater (as object)

So in this sentence:

• Amma = subject
• nýja peysu = direct object

That is why the form changes.

For the same reason: it is in the accusative singular.

The noun peysa (sweater) is feminine, and its accusative singular form is peysu.

So:

• pey sa = nominative singular
• peysu = accusative singular

Since Grandma is knitting the sweater, the sweater is the object, so accusative is used.

This phrase means for my daughter.

Here fyrir means for / on behalf of, and in this use it takes the accusative. That is why both the noun and the possessive change form:

• dóttir = daughter (nominative)
• dóttur = daughter (accusative)

And:

• mín = my (basic feminine nominative form)
• mína = my (feminine accusative singular)

So:

• fyrir dóttur mína = for my daughter

In Icelandic, possessives like mín, þín, hans, hennar can often come after the noun, especially in neutral everyday phrasing.

So:

• dóttir mín = my daughter
• dóttur mína = my daughter (accusative)

This post-noun position is very common in Icelandic and sounds natural.

English usually says my daughter, but Icelandic often prefers the equivalent of daughter my.

Í vetur usually means this winter or during the winter season.

It often refers to the winter connected to the current time frame, not just winter in general.

So depending on context, it can suggest:

• this winter
• over the winter
• during the coming/current winter season

It is a very natural time expression in Icelandic.

Certain time expressions in Icelandic simply use particular prepositions idiomatically, and with seasons í is common in expressions like this.

So you learn it as a set phrase:

• í vetur = this winter
• í sumar = this summer

Even if the preposition does not match what an English speaker might expect, this is the normal Icelandic pattern.

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

Amma | vill | prjóna | nýja peysu | fyrir dóttur mína | í vetur

That is:

• subject: Amma
• finite verb: vill
• infinitive: prjóna
• object: nýja peysu
• prepositional phrase: fyrir dóttur mína
• time expression: í vetur

This is quite close to English word order in this example, which makes the sentence relatively easy to follow.

Yes. Icelandic allows fairly flexible word order, especially with adverbials and prepositional phrases, although some orders sound more natural in a given context.

For example, you could also see:

• Amma vill í vetur prjóna nýja peysu fyrir dóttur mína.
• Í vetur vill amma prjóna nýja peysu fyrir dóttur mína.

These versions may shift the emphasis slightly, especially if í vetur is moved to the front.

But the original sentence is a very natural neutral order.

It can function as both, depending on context.

• As a common noun, amma means grandmother / grandma.
• In real family speech, it is also often used almost like a name or title: Grandma wants to knit...

In your sentence, it most naturally means Grandma as the subject.

Two important feminine nouns in the sentence are:

• amma = grandma
• peysa = sweater
• dóttir = daughter

You can often see the gender through agreement:

• nýja peysu shows feminine singular accusative agreement
• dóttur mína also shows feminine singular accusative agreement

Gender matters in Icelandic because adjectives and possessives change to match the noun’s gender, number, and case.

Yes. The finite verb vill is present tense.

The sentence means Grandma wants to knit the sweater, and í vetur gives the time frame for when that knitting is intended to happen.

So Icelandic, like English, can use present tense for a present intention about a future time:

• Amma vill ... í vetur = Grandma wants to ... this winter

It is not a future tense form; it is a present-tense sentence with a future-oriented time expression.