My Canon EOS 450D shows busy while trying to click on AF auto mode and it does not click at all.

There are several contributing factors to this problem and I'll try to list a few that may be the cause.
All Canon EOS camera auto focus modes us contrast to focus this means simply that no contrast there is no focus. There is a small green dot in the lower right corner of the viewfinder that will come on and stay on solid when auto focus as been achieved. By default the shutter will not operate "click" if auto focus has not been made.

Here are a few examples of where auto focus will ether search (focus back and forth) or fail completely.

A solid painted wall, glass between the camera and subject, mirror, chrome, any large body of water, clear sky. Another place you wouldn't expect is clear skin in your subjects face or body.

Cameras with multiple focus points and set to other then the middle will have a problem if the photographer is still attempting to use the center focus point.

Look for the green dot in the lower right corner of the viewfinder that's your focus confirmation.