Q: desktop can not connect to wifi

What airport? How old? What firmware version?

If you connect by ethernet does it work?

Desktop OS??

Do other clients work ok?

Have you done simple reboots of everything?

Was it working before some update?? Either to the OS or to the firmware of the airport??

Have you tried the wireless diagnostics??

About Wireless Diagnostics

Have you tried a reset of the Airport?

Do you have ipv6 correctly set to link-local for wireless network??

AirPort base station not seen or "An unexpected error occurred" appears in AirPort Utility 6.3