Ég þarf að kaupa klósettpappír í búðinni á horninu.

Þarf is the present tense singular form of the verb að þurfa, meaning to need.

So:

• ég þarf = I need
• hann/hún þarf = he/she needs

In other words, the 1st person singular and 3rd person singular happen to look the same here.

Here að is the infinitive marker, like English to in to buy.

So:

• að kaupa = to buy

After þurfa, Icelandic normally uses að + infinitive:

• Ég þarf að kaupa ... = I need to buy ...

Because only þarf is the conjugated verb in this sentence.
Kaupa stays in the infinitive because it comes after þarf að.

This works a lot like English:

• I need to buy
• not I need to buys

So in Icelandic:

• Ég þarf að kaupa
• not a personal form of kaupa

Because Icelandic very often makes compound nouns by joining words together.

Here:

• klósett = toilet
• pappír = paper

Together:

• klósettpappír = toilet paper

This is extremely normal in Icelandic. English sometimes writes similar things as two words, one word, or with a hyphen, but Icelandic usually prefers one compound word.

Because í can take different cases depending on meaning:

• dative for location: in / inside
• accusative for movement into

Here the store is just the location where the buying happens, so Icelandic uses the dative:

• í búðinni = in the shop/store

If the sentence were about going into the store, you would expect accusative instead.

It shows both the definite article and the case ending.

The base noun is:

• búð = shop, store

The form búðinni means roughly:

• the shop / the store in the dative singular

Icelandic usually attaches the to the end of the noun instead of using a separate word like English the.

Á horninu means on the corner or at the corner.

Again, the preposition á can take different cases:

• dative for location
• accusative for movement onto

Since the store is located there, Icelandic uses the dative:

• horninu

The base noun is:

• horn = corner

So:

• á horninu = on the corner

It describes the store:

• í búðinni á horninu = in the store on the corner

So the structure is basically:

• Ég þarf = I need
• að kaupa klósettpappír = to buy toilet paper
• í búðinni á horninu = in the store on the corner

In other words, it is the corner store that is being identified.

Because klósettpappír is being used indefinitely here, meaning just some toilet paper, not a specific known toilet paper.

That is similar to English, where we also often say:

• I need to buy toilet paper

not necessarily:

• I need to buy the toilet paper

If you meant a specific toilet paper, Icelandic could make it definite with an ending.

The sentence as given is perfectly natural, but Icelandic word order is somewhat flexible because Icelandic is a verb-second language in main clauses.

Here the subject comes first:

• Ég þarf að kaupa klósettpappír í búðinni á horninu.

But another element could come first, and then the verb would still stay in second position:

• Í búðinni á horninu þarf ég að kaupa klósettpappír.

That sounds more marked or focused, but it is grammatically normal.