Learn German: Wir bieten unserem Partner den Vertrag an.

The verb anbieten (to offer) takes two objects: the person receiving the offer, and the thing being offered. The receiver takes the dative case. Since Partner is a masculine noun, the possessive word unser gets the dative ending -em.

Why does unserem Partner come before den Vertrag?

This is a key rule for German word order: when a sentence has two noun objects, the dative noun (the receiver, unserem Partner) comes before the accusative noun (the thing, den Vertrag).

Why is the word an all the way at the end of the sentence?

The infinitive of the verb is anbieten (to offer), which is a separable verb. The core part of the verb (bieten) takes the normal second position in the sentence, while the prefix (an) is pushed to the very end.

How can we tell the gender of the word Vertrag from this sentence?

Vertrag is the thing being offered, so it is the direct object in the accusative case. Since its article is den, we know that Vertrag must be a masculine noun (der Vertrag).