I use broadband, I use a router but do I need both a router and a modem together
This is one of those "simple but once you are in .5% category it gets messy".
For most households, you will either be using a mobile modem, dsl modem inside a router, a cable modem inside router or an ONT which converts light to Ethernet!. There is also satellite.
The question you asked, is not constructed very well and is open to misinterpretation - which I and others who have replied are doing .