What does it mean when I come to a comelete stop and my car rumbles.

It rumbles as you are about to stop, well that depends.

if there is a vibration AS you are applying the brakes, and about to reach a stop, it could be that your brake rotors need machining, or "turned"; however, in this case, you would also hear a grinding, as well as feel the grinding through your foot.

there is also a "rumbling" associated with a strut mount gone bad; however, this is more of an audible clanking, and is also audible when driving slow on a bumpy road, and not really specifically in the process of stopping.

for either of these, please stop using your radio, and possibly open the windows, to better describe the sound.

IF the car shakes when sitting at a stoplight, or such, please watch your rpm gauge (tachometer), and see if the idle speed is exceptionally low or erratic.

if this is the case, it may be that your throttle positioning sensor, mass airflow sensor, or the like, is malfunctioning, and is giving a warm engine a fuel/air mix appropriate for a cold engine.