What is the difference between a router and a modem?

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I’m confused, I used to think that modem are an old thing now and nobody uses them as modems have a SIM installed in them for internet connection through ISP and it’s like a pen-drive which is insertable to a any PC like normal Pen-drives.

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Modem stands for MOdulator DEModulator. It’s a device that converts a digital electrical signal from a computer into an analog electrical signal for transmission over an analog medium, and vice versa.

That’s how we used to connect to the internet, using a modem to dial the number of an internet provider and sending data as an audio signal over the copper phone lines (dial up internet). Then we learned how to send digital data with higher frequency electrical signals over the same phone cables and called it DSL. Then we used coaxial cables used with cable TV and had cable internet via cable modems. Nowadays we connect homes via fibre optic cables, and use an Optical Network Terminal (ONT) which is basically a fibre modem (it technically doesn’t modulate/demodulate electrical signals, but serves the same purpose as a modem).

Or, we use radio waves to communicate with a nearby cell tower. For some reason we call that a cellular modem, even though it’s using the same physics as an ONT or WiFi chip.

A router translates traffic between your local network (LAN) and an external network (WAN). If that external network is the internet, the router will be talking via a modem. Some routers have a modem built into the same device. Routers will usually have other functionality like DHCP (assigning unique private network addresses to each device to let them talk to each other) or NAT (translating private address to public address so multiple devices can connect to the external network and return traffic goes to the correct device).

TL;DR A modem connects a device to the internet, a router shares that internet connection amongst multiple devices on your home network. They might be part of the same physical device. We don’t use dial up internet/modems any more, but other types of internet still use a device called a modem to establish that connection.