جير الأسنان على السن.

Because this is a nominal sentence in Arabic. In the present tense, Arabic usually leaves out the verb to be.

So:

• جير الأسنان = the subject/topic
• على السن = the predicate/comment

A natural English translation needs is, but Arabic does not:

• جير الأسنان على السن.
• literally: Dental tartar on the tooth
• naturally: Dental tartar is on the tooth.

If you wanted a past-tense sentence, then Arabic would use a verb such as كان.

Because جير الأسنان is an iḍāfa structure (a possessive/construct phrase).

In an iḍāfa:

• the first noun usually does not take الـ
• the second noun determines whether the whole phrase is definite

So:

• جير أسنان = tartar of teeth / dental tartar (indefinite)
• جير الأسنان = the tartar of the teeth / dental tartar (definite)

Even though جير has no الـ, the whole phrase is definite because الأسنان is definite.

It is an iḍāfa phrase:

• جير = tartar / lime / calculus
• الأسنان = the teeth

Together, it literally means the tartar of the teeth or more naturally dental tartar.

This is a very common Arabic pattern:

• باب البيت = the door of the house
• كتاب الطالب = the student’s book
• جير الأسنان = the teeth’s tartar / dental tartar

They are doing two different jobs in the sentence.

• جير الأسنان is a fixed expression meaning dental tartar or tooth tartar
• على السن tells you where it is: on the tooth

So the sentence is basically saying:

• Dental tartar is on the tooth

The first phrase uses teeth as part of the vocabulary item جير الأسنان, while the second phrase refers to a specific tooth, or to the tooth in a general/class sense.

If you wanted to say it is on the teeth in general, you could say:

• جير الأسنان على الأسنان

But the original sentence with singular السن is still grammatically fine.

على means on, upon, or on top of.

It is used because tartar is understood as being on the surface of the tooth.

So:

• على السن = on the tooth

A different preposition would change the meaning. For example:

• في السن would mean in the tooth, which suggests something inside the tooth rather than something on its surface.

A fully vocalized version would be:

جِيرُ الأَسْنَانِ عَلَى السِّنِّ.

Here is what the endings are doing:

• جِيرُ: nominative, because it is the subject of the nominal sentence
• الأَسْنَانِ: genitive, because it is the second part of the iḍāfa
• السِّنِّ: genitive, because it comes after the preposition على

In normal everyday writing, these short vowels are usually omitted.

A careful pronunciation would be roughly:

• جير الأسنان على السن
• jīru l-asnāni ʿalā s-sinni

A few notes:

• الأسنان is pronounced al-asnān
• السن is pronounced as-sinn, not al-sinn

That is because س is a sun letter, so the ل of الـ is absorbed in pronunciation:

• السن → as-sinn

But in الأسنان, the word begins with a hamza, so you pronounce the l:

• الأسنان → al-asnān

The sentence is divided like this:

• جير الأسنان = the subject (mubtadaʾ)
• على السن = the predicate (khabar)

So the structure is:

• [Subject] + [prepositional phrase as predicate]

This is very common in Arabic. Another example:

• الكتاب على الطاولة = The book is on the table

Your sentence follows the same pattern.

Because it comes after the preposition على.

In Arabic, nouns after prepositions are normally genitive.

So:

• على
  • السنِّ
• on
  • the tooth

Without vowel marks, you do not see the case ending in ordinary writing, but grammatically it is there.

No. It is correct, but Arabic can express the same idea in more than one way.

For example:

• جير الأسنان على السن = Dental tartar is on the tooth
• على السن جير الأسنان = On the tooth is dental tartar

The second version changes the emphasis by putting the location first.

You may also hear slightly different wording in real life, depending on context, but the grammar of the given sentence is solid and useful for learning.