Gesturinn yfirgaf hótelið.

Because Icelandic usually puts the definite article at the end of the noun.

• gestur = a guest
• gesturinn = the guest

Here, -inn is the definite article attached to a masculine singular noun in the nominative.

The dictionary form is gestur.

That is the indefinite singular form, meaning guest.
In this sentence, gesturinn is the definite form, meaning the guest.

So the pattern is:

• gestur = a guest
• gesturinn = the guest

For the same basic reason: Icelandic adds the definite article to the end of the noun.

• hótel = a hotel
• hótelið = the hotel

Because hótel is a neuter noun, its definite singular form is hótelið.

Because the sentence means the hotel, not just a hotel or hotel in a general sense.

Compare:

• Gesturinn yfirgaf hótel. = The guest left a hotel / hotel (indefinite or unnatural without context)
• Gesturinn yfirgaf hótelið. = The guest left the hotel

So -ið marks definiteness.

yfirgaf is the past tense form of the verb yfirgefa.

• yfirgefa = to leave, to abandon
• yfirgaf = left / abandoned

In this sentence, it is the singular past form used with gesturinn (the guest).

It is a correct way to say leave, but it often has a stronger sense than English leave. It can mean leave behind, abandon, or depart from.

So Gesturinn yfirgaf hótelið is natural, but it can sound a bit more forceful than simply The guest left the hotel.

If you just want to express physical departure, Icelandic often uses other expressions too, for example with fara (to go).

Because yfirgefa is a transitive verb: it directly takes something that is left.

So the structure is:

• Gesturinn = subject
• yfirgaf = verb
• hótelið = direct object

In English, this works the same way:

• The guest left the hotel

It is in the accusative, because yfirgefa normally takes an accusative object.

That is an important thing to learn in Icelandic: many verbs require a particular case for their objects.

Here:

• yfirgefa + accusative

With hótel, the nominative and accusative singular happen to look the same in the indefinite form, but the object here is still accusative in function.

• gestur is masculine
• hótel is neuter

That helps explain the article endings:

• masculine nominative singular definite: often -inn
• neuter singular definite: often -ið

So:

• gestur → gesturinn
• hótel → hótelið

Because this is the normal, straightforward Icelandic word order for a simple statement:

subject + verb + object

So:

• Gesturinn = subject
• yfirgaf = verb
• hótelið = object

That is very similar to English:

• The guest left the hotel

Icelandic can vary word order more than English in some contexts, but this sentence uses the most neutral order.

Yes, but it would mean A guest left the hotel, not The guest left the hotel.

Compare:

• Gestur yfirgaf hótelið. = A guest left the hotel
• Gesturinn yfirgaf hótelið. = The guest left the hotel

So adding the article changes the meaning from indefinite to definite.

The present tense of yfirgefa is yfirgefur in the 3rd person singular.

So:

• Gesturinn yfirgefur hótelið. = The guest leaves / is leaving the hotel

Compare:

• yfirgaf = past
• yfirgefur = present

Yes, three very important Icelandic features are packed into this short sentence:

1. The definite article is attached to the noun

   • gesturinn, hótelið

2. Nouns have gender

   • gestur is masculine
   • hótel is neuter

3. Verbs can require a specific case for their object

   • yfirgefa takes the accusative

So even though the sentence is short, it is a good example of how Icelandic grammar differs from English.