Does a router give you wi-fi?

Could someone explain to me, in simple terms, whether a router is what provides the wi-fi connection in a home or office? I've heard people talk about routers and wi-fi, but I'm not entirely clear on the distinction between the two. Is it accurate to say that a router is what actually generates the wireless signal that allows devices to connect to the internet without cables? If not, what is the specific role of a router in relation to wi-fi? I'd appreciate any clarification on this topic.