Q: enter passcode on screen comes up wrong

To use AirPlay your Mac must be running OS X Yosemite 10.10 or later

Click your Apple  top left in your screen.

From the drop down menu click: About This Mac

Which version do you see?

Good!

If you setup a password to an Apple TV, that will block AirPlay.

Check Settings > AirPlay from an Apple TV.

Otherwise try here > Use AirPlay to display video from your Mac on an HDTV

Please give the version of your Apple TV, and its software.

Please be sure you don't have Air Parrot installed on your Mac. There are reports that it conflicts with the current AirPlay.

I believe you said that you can only connect from iPhone or iPad if you use mirroring. Does that mean you cannot use normal AirPlay from those devices?

Thanks for the info.

You should check the HDMI connection from ATV to your TV. Disconnect both ends and reconnect squarely and firmly. ATV won't accept normal video AirPlay if it cannot be sure it's secure from copying. Is there anything connected between ATV and TV?

Is this a new setup, or has it been working for a while and suddenly starting giving trouble? Has anything else (router, other devices) changed recently?

Do you have other ATVs, or is this the only one?

Regarding the passcode problem, which screen does it pop up on, TV or Mac? The ATV AirPlay security will present a code on the TV, for you to enter on the computer. Is this what happens?

Have you checked the Apple support reference in Carolyn's post, especially the part about the Mac firewall?

Good. So your problem is solved, correct? Please follow up if there are any remaining issues.

Cheers