Ég skildi símann minn eftir uppi og þurfti að hlaupa aftur upp.

Yes — skilja can mean to understand, but it also has another common meaning: to leave.

In this sentence, the meaning comes from the combination skilja ... eftir, which means:

• to leave behind
• to forget somewhere and go away without it

So:

• Ég skildi símann minn eftir = I left my phone behind

This is a good example of how Icelandic often uses a verb together with a particle/adverb to create a more specific meaning, much like English leave behind, pick up, run out, etc.

Because símann is in the accusative case, not the nominative.

The verb skilja eftir takes a direct object, and direct objects are often in the accusative in Icelandic.

Here the noun is:

• sími = phone
• síminn = the phone (nominative singular)
• símann = the phone (accusative singular)

So:

• Ég skildi símann minn eftir = I left my phone behind

If you said síminn minn, that would be the subject form, not the object form.

That is the normal pattern for possessive pronouns like minn, þinn, sinn in Icelandic.

So Icelandic usually says:

• síminn minn = my phone
• literally something like the phone mine

In your sentence, because of the accusative:

• símann minn

Both the noun and the possessive pronoun have to match in gender, number, and case.

Here:

• sími is masculine singular
• the form símann is accusative singular masculine
• minn is also accusative singular masculine

Here eftir is part of the expression skilja ... eftir.

By itself, eftir often means things like:

• after
• behind
• left over

But in this sentence it helps form the meaning leave behind.

So you should learn this as a unit:

• skilja eftir = leave behind

Even though the object comes in the middle:

• skildi símann minn eftir

That split structure is very common in Icelandic.

This is a very common question, because Icelandic often distinguishes between location and movement.

• uppi = up, upstairs, in an upper place → location
• upp = up, upward, upstairs → direction/movement

So in this sentence:

• eftir uppi = the phone was upstairs / up there
• hlaupa ... upp = run up / run upstairs

In other words:

• uppi tells you where the phone was
• upp tells you where the speaker had to go

This location-vs-direction distinction is very important in Icelandic.

Because they are doing different jobs.

In:

• Ég skildi símann minn eftir uppi

  uppi describes the place where the phone was left: upstairs.

In:

• og þurfti að hlaupa aftur upp

  upp describes movement: run back up / run back upstairs.

So the sentence is basically:

• I left my phone behind upstairs
• and I had to run back upstairs

English sometimes uses upstairs for both ideas, but Icelandic often keeps the distinction clearer.

Because þurfa is followed by an infinitive with að.

• þurfa = to need / have to
• að hlaupa = to run

So:

• ég þurfti að hlaupa = I had to run

This is a very common pattern:

• Ég þarf að fara = I need to go
• Hún þurfti að vinna = She had to work
• Við þurfum að læra = We need to study

So the að here is just the normal infinitive marker, similar to English to.

Because Icelandic, like English, often leaves out the subject in the second clause when it is the same as in the first clause.

So:

• Ég skildi símann minn eftir uppi og þurfti að hlaupa aftur upp

means:

• I left my phone upstairs and (I) had to run back up

Repeating ég would also be possible in some contexts for emphasis or clarity, but it is not necessary here.

Because aftur upp is a natural way to say back up / up again.

• aftur = again / back
• upp = up

So:

• hlaupa aftur upp = run back up

Word order with short adverbs and directional words can vary somewhat, but this order is very natural. It keeps aftur closely tied to the idea of repetition/return, and upp expresses the direction.

A learner should probably remember this chunk as:

• fara aftur upp = go back up
• hlaupa aftur upp = run back up

Yes, but back is usually the more natural English choice in this context.

• aftur can mean again
• it can also mean back

Here the speaker is returning to an earlier place, so back fits best:

• I had to run back upstairs

If you say run upstairs again, the meaning is close, but back upstairs sounds more natural because it emphasizes returning to where the phone was left.

A physical phone.

• sími can historically relate to telephone, but in modern everyday usage símann minn normally means my phone as an object — for example, a mobile phone.

Since the sentence says the speaker left it upstairs and had to run back up, it clearly refers to the device itself.

Yes. This sentence is natural, but there are other ways to express the same idea.

For example:

• Ég gleymdi símanum mínum uppi og þurfti að hlaupa aftur upp.

  • literally: I forgot my phone upstairs...
  • very natural if the point is that you forgot it somewhere

• Ég skildi símann minn eftir uppi...

  • emphasizes left behind

The version with skilja eftir is especially good when the idea is I went away and left it there.

So the sentence you have is perfectly natural, and it highlights the left behind idea a bit more strongly than gleymdi.