GRE tunnel interfaces are used to facilitate encapsulation of traffic that would otherwise not be routable.

As stated by the other respondent, this may be private addresses, but it may also be non-IP traffic (IPX, AppleTalk, etc.), or multicast traffic such as dynamic routing protocols (OSPF, EIGRP, etc.).

GRE is often combined with IPSec to extend a routing domain because of IPSec's limitations.

In this case, the routing protocol (passenger protocol) is encapsulated within GRE, which is then encapsulated within IPSec (usually ESP).

GRE encapsulation results in an additional IP header (tunnel source IP, and tunnel destination IP).