Do I need extra cable to run a wireless router?
If the device in question truly is a "wireless router" rather than a wireless access point, wireless bridge, or wireless range extender, it will typically require a connection to a wide-area-network (WAN) link of some sort. In most cases this is a wired connection, but if your ISP is using carrier WiFi for the subscriber link, the WAN link may well be wireless. If you're using a carrier-provided device, it probably will come with whatever cabling is required. Most routers intended for consumer purchase come with one cable for connecting it to the upstream WAN device.
If the device is not truly a "wireless router" but is instead a wireless access point or wireless bridge, it will need cabling to connect it to a wired local area network (LAN). Wireless access points allow wireless clients to connect to an existing wired LAN, while wireless bridges allow a wired LAN to connect to an existing wireless network. A wireless access point needs cabling to attach it to the wired network that it is giving its clients access to, and a wireless bridge needs cabling to attach the devices that will use it to it.
The one situation where no cabling is required (other than power) is a wireless range extender. A wireless range extender is connected to the network only wirelessly and has no cabled connections at all.
Note that many WiFi infrastructure devices are capable of operating in any of these modes (some of them even more than one of them simultaneously), and so it's really not possible to answer your question without knowing what your actual planned use is.
Most consumers who are buying a wireless router will be connecting that router to a carrier device (called "CPE" or customer premise equipment, usually a cable modem or DSL modem); in this case you can use the cables that come with the device unless you intend to situate the router more than six feet or so from the CPE. If you what you are actually obtaining is a wireless access point to extend an existing network by adding an additional access point, you will need to run cabling from the access point back to your existing wired network elsewhere on the premises. If you're adding a range extender, you don't need cabling. Consumers almost never use wireless bridging.