The core business of a company is the business that it is primarily in.
It is generally considered good for a company to focus on its core business and get rid of other businesses unless it has strong synergies or strategic reasons to justify operating other businesses.
Companies are usually run by managers who understand the core business best (because that is what is most important) so they can generally run the core business better than non-core businesses. The latter are therefore better run as standalone businesses.
Running non-core businesses also means more a complicated management structure and distracts management from the core business.
When companies get rid of non-core businesses they may do so by selling them to a trade buyer (another company in the same industry) or by demerging them.
The problem of running non-core businesses is most acute in companies with so many diverse businesses they have no clearly identifiable core business. See conglomerates.