In case you check in manually over the counter at the airport, you must be able to present/state the airline's reference number (which you can find on your flight ticket) together with a valid ID/passport/visa. You will then receive your boarding pass and can continue through the security check and to your gate.
With some airlines, you can also check in on the machine at the airport, by providing your reference number. You will then receive your boarding pass directly from the machine, along with luggage tags that you will need to attach to the luggage you choose to check in.
Some airlines (Ryanair & Wizzair) require you to check in online and print your boarding passes before arriving at the airport. If you forget to do this, you must pay extra fees. The details on how to proceed are on your ticket.