The classic example of a multi-homed system is a machine which acts as a gateway between some LAN or intranet, and the real internet. Typically, the machine is connected to the LAN via a normal ethernet card which has an IP address specific to the LAN (one which is not recognised on the rest of the internet). It is connected to the rest of the internet via some other network device (modem, or some high-bandwidth device) which has a seperate IP address, one which is recognized as a regular internet address.
It's not technically necessary, even in the situation above, for the machine to have two seperate IP addresses - but there are various reasons why having two is an advantage. For one thing, the connection to the real internet may be via a dynamic address which changes from time to time. On the other hand, machines on the rest of the local network need to access the internet via the proxy services on the gateway machine and it is very convenient if they can do so via a fixed IP address. Otherwise, they need some method to discover the correct address, and perhaps even to change it mid-stream.
Services which run on a multi-homed host often need to differentiate between the various interfaces, seeing as they want to provide some services to the LAN only (via the LAN IP address) and others to the rest of the internet only (via the internet IP address). It's important because a service bound to the LAN address cannot be reached from the real internet and vice-versa.