It would depend on the airline and the airport/country but no matter how you buy the ticket, the next big step is what's called “checking in”. This is when the airline assigns you a seat and confirms that you will indeed be flying.
In most cases, you can now check in online a few days to a few hour before departure and print the boarding pass at home (or wherever you can find a printer). You can then show up at the airport and present this self-printed boarding pass together with your passport to drop your luggage off and go through security.
Alternatively, it's also often possible to check in at the airport itself, possibly for a fee. I have not done that in quite some time, but usually giving your name and presenting an ID was typically enough for the staff to find your booking. If using some automatic check-in machine, scanning the passport or presenting the credit card used for the booking was sometimes necessary.
In any case, you do need to be at the airport in advance (depending on the circumstances up to 2 or 3 hours before departure are recommended) but not to go there in the days before.
Finally, note that your ticket should include some confirmation number that you could write down to help the airline find your booking should any problem arise.