myers,
The console port is generally a serial connection used for directly connecting to the router for configuration. Depending on the model of the router, it may look like an RJ45 connector but it is not. It uses a special cable that connects to the serial port on your computer.
You can connect to the router's console port with the special cable but your functionality is limited to basic telnet/hyperterminal activity for configuration. You can connect to one of its ethernet ports using a straight-through patch cable (some models also have a switch that will allow a cross-over cable to be used) and use that connection to access the network/internet, etc. You will also be able to telnet over the network to the router for configuration.
Hope that helps.
The Old Man