Most networks today have some amount of in-built redundancy. However, the extent to which a network can withstand a failure varies, depending on the environment. For example, one setup might have two fully redundant paths to and from the Internet, whereas another might have Primary and Secondary firewalls, but single points of failure elsewhere.
Network devices traditionally provide redundancy at Layer 2 or 3 of the OSI model. That is, they take advantage of the existing switching or routing infrastructure to provide fault tolerance.
The principle behind hot standby router protocol (HSRP - RFC 2281) and virtual router redundancy protocol (VRRP - RFC 2338), for example, is that two or more routers share a virtual IP (VIP) address. One router takes on a primary role and "owns" the VIP; all traffic directed to the VIP routes through the primary when all is well. If the primary goes offline, a standby router is automatically promoted and takes over ownership.
Because most network devices run at Layer 2 or higher, incorporating redundancy often requires a logical topology change. As a simple example, to add the aforementioned router redundancy, you have to reconfigure all downstream routers to use with the new VIP as their default gateway.