Learn Turkish: Öğrenci kelimeyi öğreniyor. - The student is learning the word.

The base word is kelime (word). When we add the accusative suffix -i to show it is the specific direct object ('the word'), we cannot put two vowels right next to each other. We use the buffer letter -y- to connect them, making it kelimeyi.

Would the sentence still make sense without the -yi on kelime?

Yes, but the meaning would change slightly. Öğrenci kelime öğreniyor means 'The student is learning a word' or 'The student is learning vocabulary' in general. Adding the accusative suffix makes it a specific, definite object: 'The student is learning THE word.'

Öğrenci and öğreniyor look very similar. Is that a coincidence?

Not at all! They both come from the verb root öğren- (to learn). Öğrenci adds the doer/profession suffix -ci to mean 'learner' or 'student'. Öğreniyor adds the present continuous suffix -iyor to mean 'is learning'.

Why doesn't öğrenci have an accusative suffix if it translates to 'the student'?

The accusative suffix is only used for the direct object (the thing receiving the action, which is kelimeyi in this sentence). Öğrenci is the subject (the one doing the action), so it stays in its base dictionary form.