Learn German: Ich möchte den Fisch, aber mein Partner möchte den Salat. - I would like the fish, but my partner would like the salad.

In German, modal verbs (and words acting like them, such as möchten) have a special rule: the ich (I) form and the third-person singular forms (er/sie/es) are exactly the same. Since mein Partner is a third-person subject, it takes the same verb form as ich.

Why do we say den Fisch and den Salat instead of der Fisch and der Salat?

Both Fisch and Salat are masculine nouns. In this sentence, they are the direct objects because they are the things being requested. This puts them in the accusative case, which changes the masculine article from der to den.

Does the word aber change the word order in the second part of the sentence?

No, it does not. Aber (but) is a connecting word that takes up position zero in the sentence structure. The clause following it starts completely normally: the subject mein Partner is in position 1, and the verb möchte takes its usual spot in position 2.

Why is it mein Partner instead of meinen Partner?

Mein Partner is the subject of the second half of the sentence (he is the one doing the wanting), so it stays in the nominative case. Only the objects receiving the action (the fish and the salad) change to the accusative case.