Network address translation
The ISP will only allocate one IP address for a cable connection. In order for more than one computer on your network to use this IP address, there must be a device that provides what's called NAT - network address translation. This is where the router comes in, it translates the different IP addresses used on the PC's into the single allocated address from the ISP.
The router gets a different IP address for each of its connections - the cable modem side connection gets the ISP allocated address, the internal connection to which the PC's connect to is given a different address (Usually in the 192.168.0.0 range). The PC's are configured to have this internal router address as their default gateway.
Most small routers have an inbuilt switch to allow for a number of PC's to connect to it.