Hún setur armbandið á sig, en skilur ilmvatnið eftir á borðinu.

Because sig is the reflexive pronoun, meaning herself/himself/themselves depending on context.

In this sentence, the subject is hún (she), and she is putting the bracelet onto herself, so Icelandic uses:

• á sig = on herself

If you said á hana, that would usually mean onto her as a different person, not reflexively.

So:

• Hún setur armbandið á sig = She puts the bracelet on herself / She puts on the bracelet.
• Hún setur armbandið á hana = She puts the bracelet on her.
  This suggests she and her are different people.

Icelandic often expresses this idea with a verb plus a prepositional phrase:

• setja ... á sig = to put something on oneself

So the structure is very literal:

• setur = puts
• armbandið = the bracelet
• á sig = on herself

English often compresses this into puts on the bracelet, but Icelandic commonly keeps the parts separate.

The ending -ið is the definite article attached to the noun. Icelandic usually adds the to the end of the word instead of using a separate word.

So:

• armband = bracelet
• armbandið = the bracelet

The same thing happens with:

• ilmvatn = perfume
• ilmvatnið = the perfume

This is very common in Icelandic.

Because the nouns are in different grammatical cases.

• armbandið and ilmvatnið are direct objects, so they are here in the accusative
• borðinu comes after á in a location sense (on the table), so it is in the dative

This matches an important Icelandic pattern:

• á + accusative = movement onto something
• á + dative = location on something

So compare:

• á borðið = onto the table
• á borðinu = on the table

In your sentence, the perfume is already being left there, so it is on the table as a location: á borðinu.

Here eftir is part of the verb expression skilja eftir, which means:

• to leave behind
• often simply to leave

So:

• skilur = leaves
• eftir = behind

Together:

• skilur ilmvatnið eftir = leaves the perfume behind / leaves the perfume

This is similar to a phrasal verb in English. The meaning is not just the sum of skilja and eftir taken separately.

Because Icelandic often splits these verb-particle combinations.

So all of these are natural patterns:

• Hún skilur ilmvatnið eftir.
• Hún skilur eftir ilmvatnið.

In your sentence, the object ilmvatnið is placed between the verb and the particle:

• skilur ilmvatnið eftir

That is very normal in Icelandic.

Because Icelandic often omits the repeated subject when two coordinated clauses have the same subject.

So instead of saying:

• Hún setur armbandið á sig, en hún skilur ilmvatnið eftir á borðinu.

it is very natural to say:

• Hún setur armbandið á sig, en skilur ilmvatnið eftir á borðinu.

English can do something similar in certain styles, but Icelandic uses this kind of omission quite naturally.

Setur is the present tense of setja, meaning to put, to place, or in some contexts to put on.

Here it means:

• hún setur ... á sig = she puts ... on herself

So although setja often means ordinary physical placing, with á sig it commonly refers to putting on an item such as jewelry or clothing.

Yes. Ilmvatn is the normal Icelandic word for perfume.

It is made up of:

• ilmur = fragrance, scent
• vatn = water

So historically it is quite transparent, but in modern use it simply means perfume.

The verbs are:

• setur = puts
• skilur = leaves

Both are present tense. Icelandic often uses the present tense for:

• habitual actions
• simple descriptions
• narration
• textbook example sentences

So depending on context, the sentence could mean something like:

• She puts on the bracelet, but leaves the perfume on the table.
• She is putting on the bracelet, but leaving the perfume on the table.
  (less literal, more contextual)

The basic grammar is just straightforward present tense.

Yes, that is a helpful way to understand the structure.

Literally, the first clause is something like:

• She puts the bracelet onto herself

But in natural English, we usually say:

• She puts on the bracelet
• She puts the bracelet on

So the Icelandic wording is more explicit about the idea that the item is being placed on the person wearing it.

Because it has two different functions here:

1. á sig
   This means onto herself / on herself and belongs with the action of putting on the bracelet.

2. á borðinu
   This means on the table and expresses location.

So the same preposition appears twice, but it is doing two different jobs in two different phrases.

Then Icelandic would use á + accusative to show movement onto the table.

For example:

• Hún setur ilmvatnið á borðið. = She puts the perfume on the table.

But in your sentence, the meaning is location after leaving it there, so it is:

• á borðinu = on the table

This contrast is very important in Icelandic:

• á borðið = onto the table
• á borðinu = on the table

Yes. The sentence is very natural.

A learner may notice that Icelandic does not always line up word-for-word with English, but the structure here is standard:

• Hún setur armbandið á sig
• en skilur ilmvatnið eftir á borðinu

Things that are especially typical here are:

• reflexive sig
• definite article attached to the noun
• split verb-particle expression skilja ... eftir
• omission of the repeated subject in the second clause
• case change after á depending on movement vs location

So this is a very useful sentence for seeing several normal Icelandic patterns at once.