Is a router software or hardware?

wireless router is a hardware

A modern router (wireless or not) is a box that has the electronics and connections (network port connectors, radio transmitter/receivers, antenna) to allow receiving and sending of network data. It also has software (usually firmware) to allow management of the router functionality and to perform the actual routing functions.

Routers come with builtin software supplied by the manuacturer. Sometimes the manufacturer supplies updates to the router software to fix prolems in the router or to add functionality. Many routers can also be loaded with software provided from alternate none manufacturer sources to enhance its functionality or to repurpose the router entirely.