Stjórinn samþykkir fríið mitt ef ég klára verkefnið fyrir föstudag.

Because Icelandic usually attaches the definite article to the end of the noun.

• stjóri = manager / boss
• stjórinn = the manager / the boss

Here stjórinn is also in the nominative singular because it is the subject of the verb samþykkir.

Samþykkir is the present-tense, 3rd-person singular form of að samþykkja = to approve / accept.

So:

• ég samþykki = I approve
• hann/hún samþykkir = he/she approves

Since stjórinn is singular, samþykkir is the correct form.

In Icelandic, the most neutral way to say my X is often:

noun + definite article + possessive

So:

• fríið mitt = my vacation / my leave / my time off

A form like mitt frí is possible, but it often sounds more emphatic or contrastive, something like my vacation rather than someone else’s.

That is normal Icelandic structure. When a possessive comes after the noun, the noun is usually definite.

So Icelandic often says:

• bíllinn minn = my car
• húsið mitt = my house
• fríið mitt = my leave / my vacation

This is one of the places where Icelandic works differently from English.

Because mitt has to agree with frí.

The noun frí is:

• neuter
• singular

So the possessive must also be neuter singular:

• masculine: minn
• feminine: mín
• neuter: mitt

Here mitt matches fríið.

That is very common in Icelandic. Icelandic often uses the present tense for future meaning when the context already makes the future clear.

So ef ég klára verkefnið fyrir föstudag literally uses present tense, but it means:

• if I finish the project by Friday

English often does something similar in if-clauses: we say if I finish, not if I will finish.

Because after the subordinating conjunction ef = if, Icelandic normally uses subordinate-clause word order:

ef + subject + verb

So:

• ef ég klára = if I finish

This is different from main-clause word order, where Icelandic often follows a verb-second pattern.

Verkefnið is the direct object of klára, so it is in the accusative.

The dictionary form is verkefni = project / assignment / task.
With the definite article, it becomes verkefnið = the project / the assignment.

Because verkefni is a neuter noun, the nominative and accusative singular look the same, so you do not see a separate ending here.

Here fyrir föstudag means before Friday or by Friday.

The noun is föstudagur = Friday, but after fyrir in this time meaning, Icelandic uses the accusative, so it becomes föstudag.

So:

• föstudagur = dictionary form
• fyrir föstudag = before/by Friday

This is a useful pattern to remember with expressions of deadline or time limit.

Because in Icelandic, names of weekdays are normally not capitalized unless they begin the sentence.

So:

• mánudagur = Monday
• þriðjudagur = Tuesday
• föstudagur = Friday

That is different from English, where weekdays are always capitalized.

Frí can mean several related things, depending on context:

• vacation
• holiday
• time off
• leave
• day off

In this sentence, because a boss is approving it, leave or time off is probably the best fit.