Akreinin fyrir framan okkur er tóm.

It is the noun akrein plus the suffixed definite article.

• akrein = lane
• akreinin = the lane

Akrein is a feminine noun, and that matters for both the article form and the adjective later in the sentence.

That is one of the normal patterns in Icelandic. Where English usually uses a separate word (the lane), Icelandic very often adds the definite article to the end of the noun:

• akrein = lane
• akreinin = the lane

A separate definite article does exist in Icelandic, but the attached article is the usual choice in a sentence like this.

It is a fixed expression meaning in front of us. It works as a prepositional phrase, and in this sentence it modifies akreinin:

• Akreinin fyrir framan okkur = the lane in front of us

It helps identify which lane is being talked about.

Because við is the subject form, meaning we. After a preposition, Icelandic uses an oblique form instead, and that form is okkur.

So:

• við = we
• okkur = us

That is why fyrir framan okkur means in front of us, not in front of we.

It is nominative singular, because it is the subject of the sentence.

The basic structure is:

• Akreinin fyrir framan okkur = subject
• er = verb
• tóm = predicate adjective

So akreinin is in the case normally used for subjects.

Because the subject is singular: akreinin = the lane.

Er is the 3rd person singular present form of vera (to be).

Compare:

• Akreinin er tóm = The lane is empty
• Akreinarnar eru tómar = The lanes are empty

So the verb changes depending on whether the subject is singular or plural.

Because Icelandic adjectives agree with the noun in gender and number.

Since akreinin is feminine singular, the adjective must also be feminine singular:

• masculine: tómur
• feminine: tóm
• neuter: tómt

So Akreinin ... er tóm is the correct agreement pattern.

Because it belongs to the noun phrase and tells you which lane is meant.

The subject is not just Akreinin. The full subject is:

Akreinin fyrir framan okkur = the lane in front of us

This is very similar to English, where in front of us also comes after the lane.

You sometimes can move parts of a sentence in Icelandic, but the meaning or focus may shift.

In Akreinin fyrir framan okkur er tóm, the phrase clearly describes which lane.

If you said Akreinin er tóm fyrir framan okkur, it would sound more like the lane is empty in front of us, where fyrir framan okkur attaches more to the whole statement than directly to akreinin. So the original sentence is the most natural way to say the lane in front of us is empty.

Here it means something like empty / unoccupied / clear, depending on context. For a lane, it usually means there is nothing in it — for example, no car occupying it.

So even though English may choose different words depending on context, tóm is the natural Icelandic adjective here.