Learn Turkish: Biri geliyor. - Someone is coming.

Bir means "one" or "a/an". When we add the suffix -i, it forms the pronoun biri, which means "someone" or "somebody". You can think of it as literally meaning "one of them".

What is the root of the verb geliyor?

The dictionary form of the verb is gelmek (to come). We drop the -mek to get the root gel-, and then add the present continuous ending to mean "is coming".

Why is there an i before -yor in geliyor?

The present continuous suffix is -yor. Because the root gel ends in a consonant, we have to add a linking vowel before -yor so it's easier to say. According to vowel harmony rules, the e in gel tells us to use i as the linking vowel, making it geliyor.

Why doesn't geliyor have a personal ending like -um or -sun?

Because the subject here is biri (someone). This acts as a "third-person singular" subject, exactly like "he", "she", or "it". In Turkish, third-person singular verbs do not take an extra personal suffix after -iyor.