Learn Icelandic: Stjórinn þarf að lesa samninginn sinn. - The boss needs to read his own contract.

This is the key difference between reflexive and regular possessives! We use sinn (his own) because it reflects back to the subject of the sentence, Stjórinn. If the sentence used hans instead, it would mean the boss needs to read another man's contract.

How do we know to use the form sinn and not sína or sitt?

The reflexive possessive pronoun must match the noun it belongs to in gender, number, and case. It belongs to samninginn, which is a masculine singular noun. Because it is the direct object being read, it is in the accusative case. The masculine, singular, accusative form is sinn.

Why does samninginn have the definite article attached to it when we are already saying it is his contract?

Unlike in English, Icelandic usually keeps the definite article on the noun when it is followed by a possessive pronoun. So, samninginn (the contract) followed by sinn (his) translates word-for-word as the contract his, which is the standard way to say his contract in Icelandic.

Why does samninginn take the accusative case here?

It is the direct object of the verb lesa (to read). The thing that is being read usually takes the accusative case in Icelandic.