Horizontal integration is an economic term used in competition policy when you are buying units of the same kind of thing. If you own a newspaper in one city, you'll buy another newspaper in another city, and another, and another. That would be a horizontal integration.
Vertical integration is seen most particularly in broadcasting, when you own a production company that makes a program, you also own a channel, and you also own the cable system on which it is distributed. You're getting vertical integration of all the functions that have to take place to reach people.