Ég fer ekki heim fyrr en fundurinn er búinn.

Fara is the infinitive, meaning to go.
Fer is the 1st person singular present tense form, so ég fer means I go / I am going.

This verb is irregular, so the present tense does not look exactly like the infinitive.

• að fara = to go
• ég fer = I go
• þú ferð = you go
• hann/hún/það fer = he/she/it goes

So in Ég fer ekki heim..., fer matches ég.

In Icelandic, the finite verb usually comes early in the sentence, and ekki normally comes after that verb in a main clause.

So:

• Ég fer ekki heim = I am not going home

This is different from English, where we often use not before the main verb or with do:

• I do not go home
• I’m not going home

A useful pattern is:

• Subject + finite verb + ekki

Examples:

• Ég skil ekki. = I do not understand.
• Hann kemur ekki. = He is not coming.

Heim means home in the sense of going home or toward home. It is an adverb, not a noun here.

That is why there is no preposition like to.

• Ég fer heim. = I go home.
• Ég er heima. = I am at home.

This is an important distinction:

• heim = homeward, to home
• heima = at home

So Ég fer ekki heim... means I’m not going home...

In this sentence, fyrr en means until in the pattern not ... until.

So:

• Ég fer ekki heim fyrr en... = I’m not going home until...

Literally, fyrr means earlier/sooner, and en often means than. But together, in this kind of sentence, the natural English meaning is until.

You will often see this structure:

• ekki ... fyrr en ... = not ... until ...

Example:

• Hún byrjar ekki fyrr en á morgun. = She is not starting until tomorrow.

Fundur means meeting.
Fundurinn means the meeting.

Icelandic usually adds the definite article onto the end of the noun instead of putting a separate word like the in front.

So:

• fundur = a meeting / meeting
• fundurinn = the meeting

Here, the sentence is talking about a specific meeting, so fundurinn is used.

Also, fundurinn is in the nominative singular, because it is the subject of er búinn.

Búinn is an adjective/participle that often means finished, done, or completed, especially with vera (to be).

So:

• fundurinn er búinn = the meeting is finished / the meeting is over

This is a very common Icelandic way to say that something has ended or been completed.

You can also see it with people:

• Ég er búinn. = I’m finished / I’m done.
  (often said by a man)
• Ég er búin. = I’m finished / I’m done.
  (often said by a woman)

In your sentence, búinn agrees with fundurinn, which is masculine singular nominative.

Because búinn has to agree with the noun it describes.

Fundurinn is:

• masculine
• singular
• nominative

So the adjective must also be masculine singular nominative:

• fundurinn er búinn

Compare:

• málið er búið = the matter is finished
  (málið is neuter, so búið is used)
• vinnan er búin = the work is finished
  (vinnan is feminine, so búin is used)

So the ending changes depending on gender and grammar.

Icelandic often uses the present tense to talk about the future when the context makes the future meaning clear.

So:

• Ég fer ekki heim fyrr en fundurinn er búinn

can refer to a future situation even though both verbs are in the present tense.

English does something similar sometimes:

• I’m leaving when the meeting is over.

Icelandic does not have a separate future tense form like some languages do, so present tense is very commonly used for future meaning.

Yes. The part after fyrr en is a subordinate clause:

• fyrr en fundurinn er búinn

This clause explains when the speaker will not go home before.

In subordinate clauses, Icelandic word order can differ from main clauses, but here the order looks straightforward:

• fundurinn = subject
• er = verb
• búinn = complement

So the whole sentence is:

• main clause: Ég fer ekki heim
• subordinate clause: fyrr en fundurinn er búinn

A very literal version would be something like:

• I go not home sooner than the meeting is finished

But that sounds unnatural in English.

The natural English translation is:

• I’m not going home until the meeting is over.

This is a good example of why you should learn the pattern ekki ... fyrr en ... as a whole, rather than translating each word separately every time.