Some phones, depending on model/carrier, can turn off the sound in the camera settings.
If you are rooted you can delete/rename the shutter click sounds files under system/media/audio/ui.
If you are not rooted you can either mute the phone before entering the camera app (a pain in the butt). Or you can use Tasker to do it for you. You would have to program Tasker to mute the volume whenever you load the camera app. You can get more info on how to do this in the Tasker forum or just search Google as there are many explanations on how to do this.
Or use a different camera app that has a silent shutter feature.