Apple Users

Modem: Device which establishes the connection with your ISP to provide a path to the internet. These devices are typically capable of operating at a faster speed than your actual connection (metered by your ISP) speed, but it's possible if you called your ISP and ordered a higher speed, they might not notice (or know) that your modem is not capable of that speed. Have you upgraded your speed from them since you first installed the modem?

Router: Device which allows for multiple computers to be connected to one another via wires or wireless, and typically both. They are typically capable of speeds much faster than an ISP provides for the internet (i.e. are not the limiting factor).

Most devices you receive from ISP's these days are combination modem/wireless router, but it's possible to separate them. Normally you'd do this because you want a faster network in your house (communications between your various computers directly, not involving the internet), and the networking component of your ISP-provided modem/router is behind the times (G-speed rather than AC-speed for example) ... in that case you can use just the modem part of the device (to talk to internet), and get a separate, more powerful router to handle the networking in-home.