Vertical integration and horizontal integration basically refer to different processes of growth in companies.
Vertical integration is when a single companies comes to buy the entirety of their market supply chain. For this, there is backward integration and forward integration. An example of backward integration would be owning a owning a publishers and then buying an ink factory while forward integration is when a business takes control of post-production, like if a film studio bought a video store.
Horizontal integration is simply when a business wants to increase in size, start different products and/ or services. For example, when Disney bought Pixar.