Business English is English language for professionals who work in specific kinds of jobs. For example, people who work in finance, insurance, marketing or pensions will all benefit from studying this kind of English. Maybe you need to give presentations, or maybe you need to join in a conference or video call.
Conversation Class is where you can practice just speaking. In a typically English class, you study lots of grammar and lots of vocabulary. Your teacher will explain many things to you. The activities are more controlled, and in general you have less opportunity to speak. This is usually because you are concentrating on different skills in one class. For example, you can start the lesson with a listening activity, then follow this with a reading activity. After that, you may spend only the final 20 minutes of your class actually talking to your classmate.