Learn Turkish: Çocuk gülüyor. - The child is laughing.

The verb root is gül- (to laugh/smile). Because it ends in a consonant, we need a connecting vowel before the -yor ending. Since the vowel in the root is ü, 4-way vowel harmony dictates we use ü as the connecting vowel, giving us gülüyor.

Where is the word for "the" in Çocuk gülüyor?

Turkish doesn't have a direct word for "the". When a noun like çocuk is used on its own as the subject of a sentence, it is usually understood as specific ("the child"). If we wanted to say "A child is laughing", we would add the word bir (one/a) and say Bir çocuk gülüyor.

Does the root gül- mean "laugh" or "smile"?

It actually means both! Depending on the situation, gülüyor can be translated as "is laughing" or "is smiling". You just have to use context to figure out which one makes more sense.

Why isn't there a personal suffix at the very end of gülüyor?

In Turkish, the third-person singular (he, she, it, or in this case, "the child") does not take an extra personal suffix in this tense. The verb simply ends with the continuous marker -yor.