Quite often I find myself needing a large deep breath and tight chest. My mom used to have asthma. Do i? Will it effect my basketball career?

Full evaluation. You need to be fully evaluated as there are many causes for such symptoms including asthma. It would be helpful to know how long you have had these symptoms, what triggers them (playing basketball, cold exposure, house dust, etc), whether or not you smoke, and whether or not you wheeze. It's still impossible to tell what it is without a good history, physical exam and appropriate testing.
Lots of questions - . The need to take deep breaths and chest tightness can be symptoms of asthma but can also be symptoms of stress/anxiety as well as other entities. The best thing is for you to see your doctor so they can investigate. In regards to basketball, if you do have asthma it should not interfere as the medications given to you by your doctor should control your symptoms allowing you to remain competitive.