In a common Wi-Fi network, there are really only two different types of networking devices. The first is the AP, or Access Point. The second is the wireless adapter. Both come in a variety of forms.
The basic Access Point is a device that connects a wired network to a wireless one, broadcasting a signal out to client devices. Essentially, it is a network switch with a wireless card attached.
Access Point: Router
A router is an Access Point with NAT (Network Address Translator) capabilities. It is used to translate info from one network to another, usually from a private network out to the wider Internet.
Access Point: Bridge
A bridge is an access point that accepts a signal from another access point and sends it back out over a wired connection. These are used when the cost of running a wire between two parts of a building is prohibitive.
A wireless adapter is a device that plugs directly into a computer, giving it the ability to receive and transmit wireless networking signals.
Wireless Adapter: USB & PC Card
A USB wireless adapter plugs into a computer using a USB port. With higher speed wireless networks, USB can create a bandwidth bottleneck, limiting speeds. Their advantage is that they are cheaper and far easier to install. PC card adapters are similar, but usually have higher overall speed.
Wireless Adapter: PCI
These are superior to USB adapters because they connect to a computer's motherboard directly, with much higher speeds. They are commonly found in laptops with built-in wireless and on desktop computers.