I'm totally confused...

broadband is the type of connection and a wireless router lets you access that connection without a cable

Broad band refers to high speed internet service provided by your cable co. or phone company via dsl. It's sent to your house and hooked up to a modem.
Once you have the broadband (or high speed internet) run to the house and hooked up to your modem, you have to get it to your computer.
You can hook up via a cat 5 cable from your modem to your computer.
Or you can hook up a wireless router or wireless access point to the modem.
This would transmit the broadband (internet service) wirelessly to any computer that has the correct hardware to receive the signal.