Think of a router as a network traffic cop. At the very core, all routers have a main function of determining if the network data packets (i.e., the network traffic or communications) they see originating on one network (port) need to be and should be retransmitted to one of the other networks it has access to (i.e., retransmitted to one of other ports.)
Even the simplest Linksys router (now a subsidiary of Cisco) in somebody's home sees two networks. One is the internal local area network (LAN) that the home computer, (networked) printer, Xbox, TiVo, etc is hooked to and the other in the "Internet" wide-area network (WAN) as provided by the homeowner's ISP. If you print a document from your computer to the networked printer, the router knows that that traffic is all local and does not retransmit it from the LAN to the WAN. If you browse from that same computer to answers.yahoo.net, the router knows that request must be retransmitted to the WAN to be serviced, and it does so. When the response comes back, the router knows it needs to be retransmitted from the WAN back to the LAN. However, if a hacker somewhere on the WAN is trying to use Windows file sharing to see what's on the disks on your computers, the router knows (or should be able to) that did not originate as a request from your LAN and will drop or deny the request.
A Cisco router can take that idea much much further. You may have several LANs in a business setting that are kept separate. The LAN for payroll, accounts payable and receivable might be kept separate from the LAN used by product development. There's no reason for product development people to be able to see the computers and printers in HR and vice versa. There is still the need to protect all machines on the LAN from potential hackers on the WAN as before.
Additionally, the Cisco router will have much more configurability about just what shorts of traffic can be passed to and from the networks to which it is attached. A home router is mostly concerned with keeping "bad stuff" out, whereas a Cisco router might also be configured to keep the "good stuff" in. It also has extensive logging capabilities so that what data was sent where and received from where can be tracked.