Airport Extreme

Airport is behind a router of our ISP.

So, the AirPort is set up in Bridge Mode.....(Apple calls this "Off (Bridge Mode)", correct?

have setup a network, when i don't use any security everything is working and I can connect clients to the wifi and the have internet connection.

Got that.

when I enable security on the network i can no longer connect to the network, get messages like wrong password, while i know for sure that it is right.

Devices often "remember" both the name of the network and the password (if required) as part of a previous connection. If you only enable the password, then some devices will still try to use the old settings to connect to the network, which did not require a password.

To see if this might be your issue........

Change both the name of the wireless network and the password

Update to save the new settings on the AirPort

Power off the entire network.....all devices......modem, router(s), WiFi devices, printers, iPhones, etc......in any order that you want

Wait a minute

Power up the modem and let it run for a full minute

Power up the next device connected to the modem, and let it run a full minute

Keep starting devices up one at a time about a minute apart until the entire network is back up

See if devices can connect to the AirPort now

If still no luck, you will next need to reset the AirPort back to its original factory default settings and then set it up again to see if that will solve the issue.

If still no luck after that, then you likely have a defective AirPort.

I doubt that it will help, but if you have not already done so, make sure that the name of the wireless network is.......

Short.....no more than 12-16 characters. Less is better

Simple.....no special characters like an apostrophe, asterisk, etc in the name

Compact.....no blank spaces in the name

Use the same guidelines for the password

If you have tried using the standard default WPA2 Personal security settings, lower the security to WPA/WPA2 Personal.