Shutter modes are the actual speed of your shutter when you are taking your picture. When hand holding cameras you typically want to set your shutter speed to as close to just over the lens length of your camera to remove motion blur. For example, if you have a 24-105mm lens, and you are taking zoomed shots..you want to set your shutter speed to 1/125...if shooting with a 100-400mm lens, when fully zoomed, you want to set your camera shutter to 1/400.
Shooting modes are like radio dial presets. They use settings for specific situations, and put them into a single "scene". If your camera has a sports mode, it just means that your settings are probably mid level aperature, high shutter speed, mid-high iso speed. If you switch to low light mode, it will go to slow shutter speed, wide open aperature, high iso speed.
You choose manual mode when you no longer have the need for all of those modes because you already have a good sense of what settings you need in each situation..and using manual means the camera will probably be sharper, look better, and shoot faster.
Bulb just means you are going for really long exposures that you control. Instead of a set time for the shutter, you just press the shutter button to start the capture, and press it again to end the capture. Vibrations from pressing those buttons can mess up your long exposure, so some people will use an external shutter to avoid actually touching the camera.