A modem (sometimes referred to as an "aircard," "data card," or "dongle") is the device from the provider (AT&T, Verizon, etc) that provides the internet connection. With regard to mobile broadband, modems can be USB devices, ExpressCards, PCMCIA cards, or hotspots like the MiFi/Jetpack. Regardless of which form factor you choose, a modem is required in order to use the internet.
The Verizon Pantech UML290 3G/4G USB modem
A router is a networking device that you can connect your modem to in order to share the connection with other devices. If you want to share the connection from your 3G or 4G modem with multiple devices (including computers, gaming consoles, WiFi-enabled devices like the iPhone, iPad, etc), you'll need to use a 3G/4G router (“standard” routers that you may have used in the past for DSL or cable connections are NOT equipped to accept cellular modems). The router itself does not provide an internet connection - it is just a piece of hardware. While there are a few routers that have 3G or 4G modems embedded into them (like the Cradlepoint ARC Series and IBR COR routers), in most cases you need a separate modem to use in a router.
the Cradlepoint MBR1400 3G/4G router
For more helpful 3G- and 4G-related definitions, check out our 3Gstore Glossary
If you're new to mobile broadband, you might want to start here: Mobile Broadband Basics