A switch is best described as a high-performance hub that is uniquely intelligent. Data goes back and forth between the switch and during this time, the box records the MAC addresses. These addresses are unique identification numbers for hardware that is network enabled. Each sender and recipient will have different mac addresses. During this process of discovery, the hub then learns which device is linked to which port.
They can also translate multiple IP addresses on the internal network and ensure that they appear as one IP address. That IP address can then be used when accessing the larger internet network. As well as this when data arrives from the public facing IP, this is then changed into the internal IP.
Furthermore, A DNS will be part of your router and ensure that domain names are converted into IP addresses. This ensures that a router does know where outgoing traffic should be directed.
The last part of a router is a dynamic host configuration protocol. With a DHCP an IP address is given to devices on the internal network. That ensures that all devices are able to connect to the internet and IP addresses are also renewed after a short period.
More info: router vs switch
