An entrepreneur is someone who starts up a new business. It can also be used for someone who has an "entrepreneurial" spirit. One who enjoys starting new businesses and is not likely to work for someone else. An entrepreneur may also startup a business, sell it to someone else or acquire partners, and then continue to start up other businesses.
A sole proprietor is the owner of his or her own business. Sole proprietorship generally refers to the type of ownership of the business and has less to do with the personality of the person who owns it. He could have started the business from nothing (which would make them also an entrepreneur) or they could have purchased it from someone else or inherited it. Someone can own a business that they started 50 years ago.
A trader is someone who buys and sells. He might buy and sell stocks (or other securities), commodities, properties, or anything else. They are usually in the business of making profit by effectively buying and or selling things for short periods of time.