Hi....welcome to the madness!
Firstly, good advice to check out your local hobby shop (LHS) as alot of ebay stuff is junk...I'm not referring to your car though, as I don't know what it is.
Re: the battery situation...it kind of gets complicated. The current battery technology is LiPo, which are nominally rated at 7.4v...their real voltage being closer to 8.4v when full charged. The mAh rating is the capacity of the battery...ie the amount of charge it holds, or in other terms the amount of gas in the tank'. The bigger the better. Lastly the 'C' rating is the discharge rate of the battery...ie how fast the battery can supply charge. Essentially faster motors will perform better with packs with higher 'C' rating. All of this is a simplification on battery technology...it goes deeper but that's basically it in a nut shell.
You have 2 real issues though:
(1) Lipo's have been known to self combust if overly drained...when their voltage drops below approx six volts (give or take). The better ESC's have a built in feature calls a lipo cut off, which shuts the car down to prevent battery fires as the voltage approaches the cut off voltage. I'm not sure your car's electronics has this feature...if so, get a lipo. If not, get a NiMh pack, which still performs OK but is more resilient to abuse etc.
(2) If you buy a Lipo from ebay, some of the no name brands hve false ratings, so they seem cheap but you don't get what you're expecting.
Just my 2c, hope it helps, but do go down to your LHS for a better explanation.