Technically, a router is any device which routes traffic at the IP layer (layer 3; based on IP addresses like 10.1.2.3 or fe80::abcd:1234), as opposed to some other later (e.g., a device which routes at layer 2 is a switch; a device which routes at layer 7 is a load balancer).

In colloquial usage, “router” usually refers to a home internet gateway, which is a device that does a few things:

• serves as a masquerading gateway for your local network (allowing multiple devices inside your home to share a single IP address provided by your ISP)
• serves as a traditional router for IPv6 if your ISP provides such access
• contains an Ethernet switch allowing you to connect multiple Ethernet devices
• often contains a WiFi access point, allowing you to connect WiFi devices and bridge them into your main network
• has firewall/ACL systems to prevent the outside world from getting into your network

It usually does this by being a single box containing a small Linux or BSD computer running some custom software, dedicated WiFi and Ethernet chips, and some antennas. No magic there; you can do the same thing with any computer if you put enough elbow grease into it.