I assume you mean "routers" as in the stuff Linksys, Dlink, etc. sells.

Typically, when a group of computers (or other networked devices such as printers) are networked together, they are configured to communicate within its own group, called a "subnet".  If these computers or devices only wish to communicate to each other within its own group and nobody else, you won't need a router.

The primary purpose of a router is to interconnect to two different subnets together.  When Computer A needs to talk to another device on another subnet (say Computer B), it forwards that traffic to the router (Computer A does not need to know where Computer B is exactly; it only needs to know that anything other than its own subnet goes to the router).  The router will then directs the information to the other end.  Similarly, when the router receives communication that is destined to Computer A from external subnets, it will do so as well.

Hope that helps.

Another easy way to look a them in terms of home internet usage is like a phone line splitter.

They allow multiple computers to be connected both wired and wirelessly to your high speed internet provider. They also allow you to share files and printers only with other other computers connected in your house.