In business, horizontal integration is the aquisition of competitors in the same industry AND level of production while vertical integration is a style of management where companies in a supply chain are united through common ownership
These are two different types of management for controlling and expanding the business. In case of Vertical business, you try to control raw materials, production, and distribution of your products. This way you maximize profits at each successive stage by adding value. Steel and Oil industries are prime examples.
In case of horizontal business, diversification is pursued by acquiring unrelated businesses with good growth potential. In some cases competitors can also be purchased to create synergies. For example a pharmaceutical company buying another pharmaceutical company.
Horizontal and vertical can also describe how a business is structured internally, and how communication flows.
In Horizontal organization, offices at equal levels on the organizational chart can talk to each other and the communication is "flattened."
Vertical organization is hierarchical and communication must go up the organizational hierarchy and down by another path to reach another office at the same level of the organizational chart.