Learn Romanian: Profesoara ne explică că biologia studiază organismul. - The female teacher explains to us that biology studies the organism.

Yes, this repetition of sounds is called a cacophony (cacofonie). Romanian students are strictly taught to avoid it. While you will hear it in casual speech, in careful writing, a native would usually rephrase it to avoid the clash, for example by saying Profesoara ne explică faptul că... (The teacher explains to us the fact that...).

Why does the sentence use biologia with a definite article instead of just biologie?

In Romanian, when an abstract concept, a field of study, or a general category acts as the subject of a sentence, you must attach the definite article. While English just says biology, Romanian requires biologia (the biology).

Is ne a direct or indirect object pronoun in this sentence?

It is an indirect object pronoun meaning to us. The verb a explica takes an indirect object for the person receiving the explanation and a direct object, or a conjunction clause like the că clause here, for the information being explained.

Does the word organismul use a masculine or neuter definite article here?

The base noun organism is neuter. Because Romanian neuter nouns use masculine rules in the singular, it takes the standard masculine singular definite article -ul to become organismul. In the plural, it would follow feminine rules and become organismele.