The camera will display "busy" if it is still trying to achieve focus.
On the side of any auto-focusing lens (which is nearly every lens) you'll find the AF/MF switch. If the lens is swtiched to MF (manual focus) then you may find it will take a photo immediately (but it wont even try to focus.) This is a good test to see if you are running into a focus problem.
By default the camera uses "One Shot" mode which implies a feature called "Focus Priority". Focus Priority means that when you press the shutter button completely (to take the shot) the camera's priority is to first make sure that it has achieved focus and THEN take the shot after focus is achieved. If it cannot focus... it wont take a picture.
If you disalbe auto-focus then you disable "focus priroity". You can also change the camera to "AI Servo" focus mode which uses "Release Priority". Release Priority means that when you press the shutter button (aka the "shutter release button") then the camera's priority is to take the shot WHEN that button is pressed... no matter what (it will not wait for focus.) Sports photographers use this mode. But they either half-press the shutter until they've achieved focus or they enable a feature called "back button focus" (on a T2i it's assigned to the asterisk button (*) on the back of the camera ... when it's enabled.)
If the lens is having a problem with the focus motor then it would explain why your get a "busy" light and the camera wont take the picture. Do you own a different lens that you can use to test?
If you are using the camera in dark situations where you need a flash, then it would also explain this behavior. Testing the camera in brihgtly lit situations (e.g. try taking a photo outside in the middle of hte day) can help you determine if it was just a problem of inadequate light. Canon speedlite flashes such as the Speedlite 430EX II have a built-in focus-assist beam (it emits a red pattern that the camera can use to lock in focus.) The focus-assist feature can help the camera achieve accurate focus even in complete darkness.
I never heard of the 'busy' light staying on because of AF. It is possible it has something to do with your flash but way more likely you have a bad SD card. Look there first.