The router and modem are both used to connect to the Internet or a network. If we often confuse them, it is because they are mostly integrated in the box of your access provider. They have separate functions.
The modem (modulator-demodulator contraction) is the gateway to the Internet. It is responsible for translating the analog signal into computer signals (bits) usable by other devices (computer, smartphone, tablet…). It is connected to the network via the telephone jack or optical fiber depending on the chosen subscription. In principle, the modem is sufficient to access the Internet. However, if you want to connect several devices, you must add a router or a switch.
The box, a modem-router all in one
The role of the router is to distribute the traffic coming from the modem to the different access points by creating a local network, to which the different devices will connect. It connects to the modem via a Wide Area Network (WAN) outlet and distributes data via Local Area Network (LAN) outlets, assigning specific local IP addresses to each device.
Concretely, your operator’s box most often integrates both the modem and the router, as well as a switch. The latter is a kind of multiprise box with an input WAN jack and several output LAN Ethernet ports. It provides the same function as the Wi-Fi router but wired.
However, an additional router may be useful to increase the network reach, improve the connection of devices away from the box, or provide additional options. A router can also be used alone, independently of the modem, to create a private network, for example.