Tarvitsen pari uutta sukkaa.

Tarvitsen means I need.

It comes from the verb tarvita = to need.
The ending -n shows 1st person singular, so Finnish does not need a separate subject pronoun here.

• tarvitsen = I need
• tarvitset = you need
• tarvitsee = he/she/it needs

You can say Minä tarvitsen..., but minä is usually left out unless you want emphasis.

Finnish has no articles like a / an / the.

So in Tarvitsen pari uutta sukkaa, there is no word that directly corresponds to English a.
The idea of quantity is already expressed by pari = a pair / a couple.

Pari can mean:

• a pair
• a couple
• sometimes simply two, in a looser everyday sense

In this sentence, with socks, it is usually understood as a pair of new socks or two new socks.

So pari is the quantity word in the phrase.

Because after pari (and also after many numbers), Finnish uses the singular partitive for the counted noun.

So:

• uusi → uutta
• sukka → sukkaa

This is the normal pattern:

• kaksi kirjaa = two books
• kolme taloa = three houses
• pari uutta sukkaa = a pair / a couple of new socks

Even though the meaning is plural, the form after the number-like word is singular partitive.

This is one of the big differences between English and Finnish.

In English:

• two socks
• three books

In Finnish, after numbers and words like pari, the noun is usually singular partitive, not plural:

• kaksi sukkaa
• kolme kirjaa
• pari uutta sukkaa

So sukkaa is singular in form, but the whole phrase refers to more than one sock.

Because adjectives agree with the noun they describe.

Here, uutta describes sukkaa, so it must match it in:

• number
• case

Since sukkaa is singular partitive, the adjective must also be singular partitive:

• uusi sukka = a new sock
• uutta sukkaa = (some) new sock / new sock in partitive
• uudet sukat = new socks
• uusia sukkia = new socks in plural partitive

So pari uutta sukkaa is fully consistent: both words are in the same form.

In this kind of affirmative sentence, the numeral-like word pari normally stays in its basic form.

So you get:

• Tarvitsen pari uutta sukkaa.

But if the whole object has to be in the partitive, then pari changes too. For example in a negative sentence:

• En tarvitse paria uutta sukkaa. = I don't need a pair/couple of new socks.

So pari is not always fixed; it depends on the grammar of the whole phrase.

Yes, and the meaning changes slightly.

• Tarvitsen pari uutta sukkaa.
  = I need a pair / a couple of new socks.

• Tarvitsen uusia sukkia.
  = I need some new socks.

The second version is less exact about quantity. It sounds more indefinite.

Yes, but that is not exactly the same.

Tarvitsen uudet sukat can mean something like:

• I need the new socks, depending on context
• or I need new socks / a new pair of socks, if the situation makes that clear

Compared with that:

• pari uutta sukkaa explicitly gives a quantity
• uudet sukat sounds more like a complete set or just new socks in a more general way

So the original sentence is more specific about number.

Inside this noun phrase, the normal order is:

quantity word + adjective + noun

So:

• pari uutta sukkaa
• kaksi isoa taloa
• kolme hyvää kirjaa

The adjective normally comes before the noun it describes, and the number/quantity word comes before both. So uutta pari sukkaa would not be standard Finnish.