Audio Systems: Car Stereo Problem, guage wire, amp fuse

Question
Hello,
         When I connect my car stereo to the car, it switches on
fine. But when I put it on cd player mode, it starts fuzzing then
just cuts out, I know it is not due to the stereo because I tried it
in someone else's car and it works fine. There must be a lack of
power somewhere, is there an extra cable I can buy to give it
more power, to stop it from cutting out.

Thanks for your time.

Answer
sure you can run your own wires, I dont know if that's the problem or not.  The amplifier portion of the stereo will draw more than the little motors that the cd player uses (and the radio doesnt)  I would try turning up the radio real loud and see if it goes staticky as well.  if so, then I agree that you have a power problem.
regardless....  if you want help running new power there are a few ways to do this,  I think the best is to physically pull a posative wire from the battery to the location of the cd player, attach it to the battery using a ring terminal on the end of the wire. (12 guage wire is plenty) make sure it's fused with about a 10-20 amp fuse very near the battery (8-12 inches). Be carefull working with car batteries the power inside them is very dangerous. it also helps alot to run your own negative/ground.  use the same gauge wire (12 guage) and find a good solid metal part of the car as close to the cd player as possible, use sand paper to remove all the paint and shine it up bright about 1 inch in diameter, drill a hole in the center of where you shined it up (make sure you dont hit wires, break lines, fuel lines, etc).  crimp a ring terminal onto the ground wire and screw it down through the hole you just drilled into the car. it's best to use machine screws and put a nut on from the bottom to get it extra tight but if that isnt avilable use a good size sheatmetal screw to screw down the ring for the ground.  I like to spray paint over the whole thing so it doesnt rust and corrode.  
Now you have a good strong constant posative and ground. Most stereos have 2 posative wires (usually a red and a yellow) one is for constant power (usually the yellow) and one is for switched power (usually the red). Depending on the model of stereo the yellow may provide just memory backup to retain your settings and the red provides the majority of the power, or the yellow may provide the majority of the power and the red only triggers the unit to turn on.  The only way to know how for sure is to use a meter that can measure current and I highly suggest getting one.  Mine was 18 bucks and works great so you dont need to spend a fortune.  
you can try attaching the new ground to the stereo's ground and the new postaive to the constant posative of the stereo (usually the yellow). leave the red hooked up the way it was to the car.  try the stereo out and if this solves the problem you can consider yourself done.  If the problem still exists then try attaching the both the red and yellow to the new wire and try the stereo.  in this wiring config you are providing plenty of power to the stereo and if there is still a problem you can rule out power.  Note that you can't leave it hooked up this way since the stereo is constantly triggered to turn on and will drain your battery, its just ment as a test.
If this way does fix your problem then we need to also beef up the current for that red wire and I can help with that but that can be more complicated so try all of this stuff first and let me know what happens.

GOOD LUCK!!!