Learn Croatian: Razgovaram s profesorom. - I am talking with the professor.

The preposition s (meaning 'with' a person) requires the noun following it to be in the Instrumental case. For regular masculine nouns that end in a consonant, like profesor, you add -om to make it Instrumental.

Sometimes masculine nouns take -em in the Instrumental. Why doesn't profesor do that?

Masculine nouns only take -em if they end in a 'soft' consonant, such as c, č, ć, đ, j, lj, nj, š, or ž. Because profesor ends in 'r', which is considered a 'hard' consonant, it gets the regular -om ending.

Why is the word for 'with' just s here instead of sa?

The standard word for 'with' is s. You only add the 'a' to make it sa if the following word starts with a sound that is hard to say right after an 's', like s, z, š, or ž. Since profesorom starts with a 'p', it is easy to pronounce s profesorom, so the extra 'a' is not needed.

Could I use govorim instead of razgovaram to say I am talking with the professor?

You could, but razgovaram is a better fit here. The verb govoriti generally means 'to speak' in a one-way sense (like giving a speech or speaking a language), while razgovarati means 'to converse' or 'to have a conversation'. Since you are talking with the professor, it implies a two-way exchange.