Easiest way is to read the signs.
In small airport there is only a small set of check in counters, just read the sign behind each to determine which airline is checking in there.
In larger airports there is either a electronic signboard telling you what row or area each airline is located or signs outside the various entrance doors to indicate which airline inside.
The departure gates are in a secure area and you can't go there until after you have checked in.
About the only time you would check in at the departure gate would be if you are flying in from another airport on another ticket.
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The first thing to do is go to the departure screens, which list not the departure gates, but the check-in rows, or aisles, for all airlines.
You go there, look for the airline (sometimes the same aisle can serve two airlines), and wait in line.
The departure gate is where you will board the plane.
You'll see screens with the flight numbers and gates, after security and immigration checks.
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Check-in is before the serious security check (you won't get through security without a boarding pass).
There will be signs showing you where to check-in for a particular airline.
Departure gate is after security.
It's where you go to actually get on the plane.
There will be more signs showing you were the various gates are.
Notes
If you don't have hold luggage, you often don't go anywhere tocheck-in.
You do that online, before you get to the airport andeither print off the boarding pass, or get a code on the airline'sapp on your phone.
If you haven't checked in online, at leastRyanair will let you check in at the airport - but they will chargeyou extra for it (so remember to do it online).
Some airports have a lightweight security check for everyone on entry to thebuilding.
This is the not the main security check before entry to the secured "airside".
Do not be confused by it.
travel.stackexchange.com/users/52986/Martin Bonner supports Monica
Your ticket, boarding pass or confirmation mail will tell you which terminal to go to (if there's more than one).
Once you're there, you read the signs and information screens.
travel.stackexchange.com/users/11491/David Richerby
once you are in the terminal you can: approach a information point, they will provide you the info about which counter is the airline and which gate you have to proceed. or by huge cities or capitals you can see the issue that the airport has more than one Terminal, in such cases the Airline is always giving you that info as soon as you do the check in... they will have a huge departure time table...
there you can spot your flight
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In Germany, large airports whose terminals are somewhat distant (FRA, MUC) even show on the roadsigns near the airport which airline is located at which terminal.
Good luck reading those at 80 km/h, though :)
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Out of interest, how do you know where you are to go once at the airport to check in and what is it called?   Is it departure gate on the board you go to with the flight number and airline which tells you what gate? And is that where you would check in and get the boarding pass from to go through security etc?
How do you know where to go to check in at airport/airline?
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Your ticket, boarding pass or confirmation mail will tell you which terminal to go to (if there's more than one). Once there, you can check the departure screens or ask at the information desk for the correct check-in row or aisle for your airline. Once you have checked in, there will be other signs showing you where the gates are. Navigation to your check in station and departure gate are both done with the help of screen found inside the airport. You will also receive a confirmation mail that will tell you, which terminal to go to at the airport before you arrive. The check-in station is where you deposite your bag and the departure gate is where you get on your plane. If you encounter any problems you can make an enquiry at an information point at the airport.
In most cases airports have signs or screens that provide the location of your airline check-in station and once you go passed security there will be more screens telling you which gate you departe from You should receive a confirmation mail that will tell you where to go at the air port There are information points at the air port that you can ask about your flight and any other quiries you may have.