A network is typically composed of the following hardware components:
There are actually two separate networks shown on the above diagram: the local network, consisting of all the devices pictured; and the internet. The router acts as the link between the two, providing all the devices on the local network with internet access.
Wireless acess point (WAP)
This device adds wireless capability to the local network. It acts as a "bridge" between the wired and wireless parts of the local network, connecting all devices together on the same network.
If you are using an AirPort Extreme or Time Capsule for this function, in addition to a router, these devices must be set to "bridge mode" to perform the function of a WAP, otherwise they will act as routers in their own right and will make configuring the wireless part of the network very difficult. This applies to any wireless router that you want to use as a WAP.
Many routers have wireless functionality built in, so there is no need for a separate WAP.
These are the devices that connect to the local network, either wired or wirelessly. Each device can communicate directly with any other device on the same local network, but in order to communicate with any device over the internet, it must go through the router.