A business is an organization intended to make profits, i.e. to make money for agents who own it. Note that not all commercial organizations are businesses. For example, in a franchise, while the parent company is regarded as a business, the sub-organizations are commercial organizations, but not businesses. Also, businesses are typically legal agents, while commercial organizations need not be.
A commercial activity is a kind of economic action and a kind of transaction. Commercial activities include all types of buying, selling, offering to sell, offering to buy, marketing, paying for bought items, requesting bids, performing services for hire, advertising, etc. Its many types include repaying loans, renting, and recalling products. Though there is a great deal of overlap between commercial activities and business events, they are not the same. Firstly, business events do not include any purely social or recreational events (whereas a weekend swap-meet, for example, would be considered a commercial activity). Secondly, not all activities carried out in pursuit of business interests fall under the definition of commercial activity above (consider head-hunting, for example).