Learn Turkish: Başkan plan yapmıyor. - The president is not making a plan.

No, we would ask Ne yapmıyor? (What is he not making?). Because plan doesn't have the specific accusative ending, it is a general object ("a plan"), not a specific one ("the plan"). To answer the specific question Neyi yapmıyor?, the sentence would need to be Başkan planı yapmıyor.

The translation says "a plan", but there is no bir in the Turkish sentence. Why?

In Turkish, when a noun is right next to the verb and has no endings, it acts like a general category. Plan yapmak is almost like the single concept of "plan-making". You can say Başkan bir plan yapmıyor, but dropping bir is very natural for general actions.

Why is the negative part of yapmıyor spelled with an ı instead of an a?

The basic negative suffix is -ma or -me (so, yapma). However, the present continuous suffix -yor changes the vowel right before it. The a in -ma "narrows" to an ı to flow better with the word, making it yapmıyor.

How do we know it means "The president" and not "A president" if there is no word for "the"?