Ég horfi í gegnum gluggann.

Horfa is the infinitive, meaning to look or to watch.

In the sentence, the verb has to match ég (I), so it changes to the 1st person singular present tense:

• ég horfi = I look / I am looking
• þú horfir = you look
• hann/hún/það horfir = he/she/it looks

So Ég horfi ... is the correct finite verb form for I.

It can mean either, depending on context.

In Icelandic, the simple present often covers both:

• I look through the window
• I am looking through the window

If you want to emphasize that the action is happening right now, Icelandic can also use vera að + infinitive:

• Ég er að horfa í gegnum gluggann.

But the shorter sentence Ég horfi í gegnum gluggann is perfectly normal.

Í gegnum means through.

It is a very common Icelandic expression and functions much like a preposition in English. In this sentence, it introduces the thing that someone is looking through:

• í gegnum gluggann = through the window

It is normally written as two words: í gegnum.

The basic noun is gluggi, which means window.

Here it appears as gluggann because:

1. it is definite = the window
2. it is in the accusative singular

So:

• gluggi = window
• gluggann = the window

Unlike English, Icelandic usually adds the as an ending on the noun rather than using a separate word.

Because í gegnum takes the accusative.

So the noun after it has to be in accusative form:

• glugga = a window
• gluggann = the window

This is something learners often just have to memorize together with the expression:

• í gegnum + accusative

The preposition changes the meaning quite a lot.

• horfa á gluggann = look at the window
• horfa í gegnum gluggann = look through the window

So horfa á focuses on the object you are looking at, while horfa í gegnum focuses on looking through something.

It can be correct, but there is a small nuance.

• horfa í gegnum gluggann = look through the window
• horfa út um gluggann = look out of the window / look out the window

If the idea is simply that someone is gazing outside, Icelandic very often uses út um gluggann.

If the idea is more literally through the window, then í gegnum gluggann works well.

As a learner, you should normally keep it.

Icelandic verbs do show person and number, but subject pronouns are still commonly expressed in ordinary sentences. So the neutral, standard form is:

• Ég horfi í gegnum gluggann.

Leaving out ég is not the normal default in the way it is in some other languages.

Yes, but Ég horfi í gegnum gluggann is the normal, neutral word order.

Icelandic does allow other orders for emphasis. For example, you could put the prepositional phrase first:

• Í gegnum gluggann horfi ég.

That sounds more marked or literary, because it highlights through the window.

A useful rule for learners is: start with the standard order unless you have a reason to emphasize something.

Then you would use the indefinite accusative form:

• Ég horfi í gegnum glugga. = I look through a window.

Compare:

• glugga = a window
• gluggann = the window

So the ending changes depending on whether the noun is indefinite or definite.