Switches (particularly layer-3 switches) have basic routing capabilities to connect different VLANs, but that is not their focus. Their purpose is to facilitate communication between devices connected to the same subnet, and across subnets on the same LAN.
Routers specialize in communication between networks, e.g. between a LAN and the internet. They can use static routes or dynamic routnig algorithms protocols (e.g. RIP, EIGRP, OSPF, BGP) to find the shortest route, often across many routers, from the source to the destination.
Think of routers as intercity railway lines, and switches as local transportation.
The device that is usually referred to as a "home router" is usually a combination of a router, a switch, a wireless access point, optionally a cable modem, and sometimes a telephone modem; plus it offers services like a firewall, NAT, and sometimes VPNs. It does everything, but with a much lower performance compared to dedicated hardware.