Learn Turkish: Kedinin tüyleri kirli. Onu yıkar mısın? - The cat's fur is dirty. Will you wash it?

In Turkish, the aorist (broad) tense is the standard way to make polite requests. While it translates to "Will you...?" in English, using the actual future tense (yıkayacak mısın?) would sound more like asking for a factual prediction ("Are you going to wash it?") rather than asking for a favor.

How exactly is the verb yıkar mısın formed?

It comes from yıkamak (to wash). Since the verb stem yıka- ends in a vowel, we just add -r to make the aorist form yıkar. Then we add the question particle and the personal ending for "you" (singular/informal): mı + sın.

Why does it say Onu instead of just O?

Onu is the accusative (specific direct object) form of O (it). Because you are asking someone to wash a specific thing (the cat), the pronoun takes the accusative case ending -u. The -n- is a buffer letter always used when adding suffixes to pronouns like O.

Doesn't tüy mean feather? Why is it translated as fur?

You are right that tüy means feather for birds! However, in Turkish, tüy is also the standard word used for the body hair or fur of animals like cats and dogs. The plural tüyleri literally means "its hairs", which translates naturally to fur in English.