A router is a networking device whose software and hardware are usually tailored to the tasks of routing and forwarding information. Routers connect two or more logical subnets, which do not necessarily map one-to-one to the physical interfaces of the router. (All this information can also be found on Wikipedia with a lot more details)

From an Eye-Fi perspective, you need a wireless router to transfer photos from the Eye-Fi Card to your computer (PC or Mac). When transferring photos to your computer, the card connects to your wireless router and sends the photos to a computer (PC or Mac),running the Eye-Fi Manager software, that is designated as the destination for the photos.