Is my finger broken or fractured a pool ball hit my left index finger and it swelled up twice its normal size and I get sharp shooting pains and its bruised

Broken. Broken and fractured are two ways of describing the same thing. Of course there are many ways to describe breaks/fractures i.E open or closed, displaced or non-displaced (in or out of place), articular or non-articular (involving the joint or not), comminuted or non-comminuted (clean or not clean) etc..
You. You could have a fracture. Based upon the injury it may be a non displaced tuft fracture or a fracture of the distal phalanx. Another possibility and the more like;y one is that based upon mechanism it is a severe bruise. An xray will help define the injury. The treatment may not differ much but it will allow you t know how long to expect it to hurt. Make sure that if there is a nail laceration or bleeding below the nail tat the finger is kept clean.
It. It is not possible to tell if your finger is fractured / broken with this information. I recommend that you see a doctor and have it x-rayed.