Messages in some decentralized networks (Gnutella is one example) move by hopping from node to node. In order to keep messages from traveling forever, messages expire after a certain number of hops. The horizon is the furthest distance (as measured in hops) a message may travel before expiring. It is called the horizon because it is the furthest extent of a node's view of the network.
A message starts out with a number of hops it may travel before expiring, called the time to live, or TTL. At each hop the TTL is decremented. When TTL is 0 the horizon has been reached.