Ok - it depends :p
If you want an internet connection, you'll need to pay for it. Strictly speaking, that connection ends at the plug at the building - be it DSL, Cable, fiber or anything else.
Next step is you'll need a box that goes into that plug and converts whatever is running through the plug into Ethernet (and occasionally a couple of phone lines). Usually, that box is a modem, although it may be something else in special cases (e.g. FTTH). Depending on where you live, that modem will be supplied by your ISP (Internet Service Provider) or you can supply your own (but have to configure it according to the instructions from your ISP). So, depending on this, the modem may be included in your monthly internet fee, rented seperately or paid by yourself.
Since most modems are pretty simple on the ethernet end, you'll also need a router after that. That router will provide you with multiple ethernet ports and a (one) WiFi access point. It might be combined with the modem or not, and you might wish to add one yourself in any case (when e.g. the WiFi part of the modem from your ISP doesn't do what you want to do). So, depending on what your ISP provides and what you actually want, one WiFi access point may be included in your monthly fee to the ISP (it usually is) or not.
And finally, if you want to run more than one access point (e.g. because the one on the router is too far away/too weak to reach your favourite seat on the porch), you're completely on your own, i.e. you need to buy the corresponding hardware and set it up (or pay or otherwise entice someone else to do so).
So, you pay a monthly fee to your ISP for your internet connection. That fee may or may not include a router which may or may not supply WiFi to your home which may or may not suffice for your needs.
For private users, the _usual_ configuration would be that your monthly paid service includes a box with a combined modem/router/WiFi access point that is to a large part remotely administered by your ISP.