Learn Arabic (Modern Standard): إطار السيارة هنا.

In an Arabic possessive phrase (Idafa), the first word is the thing being owned. A strict rule of this structure is that the first word never takes ال (al-). It becomes specific automatically just by belonging to the car.

Is there a word for 'of' or an apostrophe-s to show possession here?

No, Arabic shows possession simply by placing the two nouns right next to each other. إطار (tire) followed immediately by السيارة (the car) automatically translates to 'the tire of the car' or 'the car's tire'.

If إطار doesn't have ال, how do we know it means 'the tire' and not 'a tire'?

The whole phrase gets its 'the' or 'a' status from the second word. Because السيارة has ال (the), the entire phrase is specific: 'the car's tire'. If you wanted to say 'a car's tire', you would remove ال from the second word, making it إطار سيارة.

Where is the verb 'is' in this sentence?

Arabic doesn't use a present tense word for 'is' or 'are'. The sentence is literally 'Tire (of) the car here'. You simply state the subject, إطار السيارة, and follow it with the information about it, هنا (here).