Learn Turkish: O doktor, biz öğrenciyiz. - He is a doctor, we are students.

The word öğrenci (student) ends in a vowel. To say "we are", we use the suffix -iz. Turkish doesn't like two vowels next to each other, so we add the buffer letter -y- between them, making it öğrenciyiz.

Why doesn't doktor have a suffix to mean "is"?

In Turkish, the third-person singular (he/she/it is) usually has no visible suffix. Just saying O doktor is the complete and correct way to say "He is a doctor" or "She is a doctor."

English uses the plural "students," so why isn't there a -ler plural suffix on öğrenci?

Because the "we are" suffix (-yiz) already makes it perfectly clear that there is more than one person, adding the regular plural suffix -ler is not necessary. Biz öğrenciyiz sounds perfectly natural for "We are students."

Can I drop the pronouns o and biz from this sentence?

You can easily drop biz because the -yiz suffix already tells us the subject is "we." But if you drop o from O doktor, it might be confusing who you are talking about, since there is no ending on doktor to give us a clue!