Broadway Blue: Heres a brief explanation,,say your size is 195/60/15 (which you said yours are)
This means the wheel diameter is 15"
the tread width is 195mm
the "aspect ratio" of the sidewall is 60 (Im not sure if thats % but possibly) i think 60% of the width...this ones a little tricky
the lower the aspect ratio (ie 50, 40) the narrower the sidewall of the tire (looks more like a rubber band around the wheel) "low profile" is what this is also known as...
You can get much better performance (if thats what you are looking for) by going to a wider tire with lower aspect ratio (ex from 195/60 to 215/50)lower profile tire...increasing rim diameter also helps (aka plus 1 and plus2 - going up to 16 or 17 inch wheels)
the tire is wider, the sidewall narrower, putting more tread on the road , GENERALLY resulting in better cornering and braking..some combinations can hurt performance though. Also, going to a lower profile tire increases stiffness and causes a stiff (possibly harsh) ride dpeending on how stiff the ride already is...
From my personal experience, i ALWAYS like to get a tire that is somewhat more performance oriented than the stock size...example: in my first beat-up car, i went from a narrow, tall 185/80/13 to 195/70/13's , which were significantly wider and handled MUCH better..however, 195/60 is fairly low profile to begin with, and 50 and 40 series tires are high performance tires not ususally suited to your type of car...
if you are looking for the same ride youve always had, long life and low cost, stick with the size youve got...if you are looking for better performanace without getting larger wheels, try a 205/55, 215/55 or 225/50 tire...you will get a stiffer ride but better cornering...