A wireless router is a device that allows multiple devices to connect to a local area network. Wired routers are becoming a thing of the past. Most homes and businesses are going completely wireless.
Take a modern home for example. Today we have not only computers that are connected to our networks. We have our phones, laptops, gaming devices such as Xbox 360, and even our televisions. Not only are these things connected to one router, they are connected without the need of wires.
Before the wireless router you had to have all of these things in the same room, or you would have to run wires through your walls and around furniture in order to connect them all to the router. If you were lucky enough and wealthy enough you could have your home wired while it was being built, but that was not available for the average home owner, so instead you would see wires running from one end of the living room to the other.
But, with the advent of the wireless router you can now connect all of these devices throughout your home with the literal push of a button. There is a small button on each of these devices that allows the router and the device to communicate automatically and connect with each other.
This is typically a one-time task that would need to be done. If you turn one of your devices off to move it for example moving the printer from one room to another you may have to reconnect with the router, but with the ease of use that these newer devices have it is typically not a big deal.
Before we had wireless routers the task of moving equipment from one room to another was such a painful and confusing experience that it was not done very often, and typically only due to necessity.