Broadband refers to the internet connection supplied by an internet service provider (ISP), which is usually done by passing data through some sort of cable to your router, whereas Wi-Fi refers to the technology which allows you to wirelessly connect a device to your router in order to access this data. You can also connect a device to your router more reliably using an ethernet cable, but the convenience of Wi-Fi technology has meant that most people now only connect to the internet wirelessly.