In television, the major U.S. broadcasting networks are ABC, CBS, Fox, NBC and PBS. The network airs programs that run on all of its stations -- like American Idol, which airs on the Fox network nationally in prime time.
A network may own a local station outright or have an affiliation agreement with a company that owns a station. That agreement is a contract binding the local station to the network for a set number of years. Occasionally, a station will switch affiliations.
In the mid 1990s, Fox was able to get several local stations in major cities to switch from the "big three" networks -- ABC, CBS or NBC -- to Fox. That gave Fox's Nielsen ratings a huge boost and helped it achieve parity with the other networks.
In cable television, some channels use the word "network" in their name even though they are a single channel and do not meet the definition of a network. The Food Network and the Game Show Network are two examples.
CNN's official name is the Cable News Network. While it is a cable channel, it does have agreements with many local stations across the country to share news stories and video, which makes it similar to a network. Those stations sign contracts with CNN to share resources, even though the stations are also affiliates of one of the broadcasting networks.
Stations do that to double their news resources. A station that's an affiliate of both CBS and CNN can use either source as it sees fit. CNN may have better video of a tornado touching down than CBS, so the station may choose to air CNN's video.