This is dependent on the airline and airport.

For some airlines, it’s still just a ticket, and using your passport or other ID you will be able to check-in (either at a desk or at a kiosk) to get your boarding pass. When using a kiosk, you may be able to use other ways to retrieve the file: PNR locator (the booking reference, usually a 6-letter code), frequent flier card or number, etc. This varies quite a bit. But as you’ll often need ID at some point anyway...

For others, the “ticket” is actually directly your boarding pass, and they usually need the barcode that is printed on that ticket. Some airports may accept scanning it off a mobile (most do nowadays, but there’s probably the odd exception), all with accept the paper version. Many LCCs will charge you to get the boarding pass printed at the airport if you didn’t print it in advance (or don’t have the barcode on your phone, if that’s accepted).

Just your e-Passport.

AFAIK: the smoothest process I've ever had was just scanning my e-Passport at the self-service check-ins. This was with British Airways in Gatwick but I've seen similar at many airports.

You usually only need the passport at gates too, the boarding pass is only needed for passing security and shopping in the duty free, most of the times.

However, if you're not required to prove your ID, like when travelling in the Nordics (where I live) you might not have an ID on you...

Another common way is using only the ticket number.

When checking in manually (at the check-in disk) you only need you passport too. They can check if you're on the passenger list and issue you your boarding pass.

Low Cost Airlines

Being used to the above I thought the same applied to low cost airlines (Wizz Air for example), I had to pay 60 € extra because I hadn't saved the boarding pass PDFs 2 hours before departure, nor printed them 2 hours before. These things are different between airlines and I therefore consider this question very broad.

In the browser in which you can access the ticket, hit the print button, and print to a PDF file. Then send that PDF to your phone, in a way that you're sure it gets cached!

If you don’t have something the scanner will accept, you have to get an agent or kiosk to print one. This requires a passport or other photo ID, or your name and six-character confirmation, or the card used to buy the ticket.

I always print mine in advance, because more than once, the scanner has not been able to handle an image on my cellphone. And with one exception, the kiosks have always failed to scan my passport.