American Airlines has suspended it's cancellation fees, but does this mean that I get the money back for my ticket also?

No. what that means, is they will issue you a credit for the full amount that you purchased your airfare without charging the change/cancellation fee. You then will have a year to use/purchase new airfare using the credit.

For example, if they issue you a credit on March 23, 2020, you have until March 22, 2021 to use the credit to purchase airfare. You do not have to fly by March 22nd 2021, just purchase so you can use the credit to buy airfare for Xmas 2021.

MAYBE

It might be just a return of ca

sh or credit for a new flight.

There is not ONE ANSWER for every possible ticket situation.

There is a reason that American Airlines has published  the procedures on what they can do for you.You are quite capable of reading it  yourself.

If your flight has been canceled, we'll send you an email with more information. The email will include your ticket number. Keep this handy so you can refer to it when you call to rebook.

https://www.aa.com/i18n/travel-info/coronavirus-up...

You may not get a refund.  Only the airline can tell you for sure.  This just means that you won't lose any of the value of your ticket to a cancellation/change fee.  It's likely that you'll just keep the full value as credit toward a future flight.  Again though, you need to contact the airline instead of just believing the one person who told you what you want to hear. 

You have gotten different answers from people here, so obviously we do not know.

CALL the airline and ask them. They are the ones making the rules.

Suspension of cancellation fees means that you won't lose the value of your ticket. If it's a non-refundable ticket, you won't get a refund UNLESS American Airlines canceled the flight. Otherwise, you'll get the value of your ticket towards a new ticket 

Yes. That's what the suspension of cancellation fees means. They're not going to charge you if you cancel. You'll get a full refund.

No idea. Call the 800 number and ask them. Every ticket is different.