A typical WiFi setup consists of a broadband (cable or DSL) modem connected to a wireless (WiFi) router. One or more computers can be connected to the router by Ethernet or by wireless adapter(s). [pictorial overview]
The router establishes the wireless LAN (local area network) required to let connected devices share a single Internet connection. Inside a WiFi router, you'll find a small computer, a digital radio to communicate with devices, flash memory to retain settings, and programs to run everything.
WiFi routers provide a digital radio "home base" for WiFi enabled computers to connect with. This radio is used to establish an "access point" that multiple devices can connect with. Each device connects using it's own digital radio, called a wireless adapter. Most WiFi routers also provide Ethernet ports for wired connections.
The router must pass on and keep track of Internet requests made by each device, and direct each Internet response back to the originating device. It does that by giving each device a different LAN address. The router uses the addresses to identify which data packets belong to which devices. The router may also incorporate a state-inspection firewall.
Wireless, or WiFi adapters are used to connect devices to wireless networks. Adapters interconnect with the device's processor internally or via a plug-in arrangement. Most newer laptops already have internal WiFi adapters. "WiFi Ready" devices like GFI sets, printers, televisions, game consoles, and home theater equipment also come with internal adapters. Coffee pots and toasters are probably not far behind.
Many other wireless devices -- microwave ovens, cordless phones, Bluetooth, wireless video cameras, outdoor microwave links, wireless game controllers, etc., operate in the same frequency bands as Routers. You don't of your router (and laptop) too close to any of these interference sources.
It probably isn't critical where you place your router if 50 feet or so is the longest range you need to reach. Obstructions can reduce that range though. For example, I have a room with a mirrored wall. Even though the distance is only 25 feet, the signal is not very strong on the other side of that silvered (actually aluminum) surface. Walls, plumbing, wiring, etc. also attenuate the signal.
Of course you want to be able to place your router near your modem, but if you need to place your router more centrally you can use an Ethernet cable to separate them. Sort of defeats the wireless idea, but it's one solution. Otherwise you'll need a phone jack or cable outlet that is more centrally located.
Placing a router is much like using a cell phone (except WiFi radios aren't as nearly as powerful). You may need to move things around until you get enough "bars" to make your WiFi work. ;-)
The router handles communication between your ISP (Internet service provider) and your LAN (local-area network). ISP networks are a type of WAN (wide area network). The router appears as a computer to you modem. Devices, like computers, "connect" to the router either wirelessly, using a "wireless adapter", or directly with Ethernet cables. [configure]
All the computers or other devices that "connect" to the router can access the Internet in much the same way they could if they were connected directly to the modem. In addition, they can communicate with each other, if you set them up with additional capability.