Basically I have a few questions. I have something that takes 120 watt and 12 volts. Normally plugged into a car to work. (Its a a specific Light I use). Is it possible to buy a battery in which I can connect and run this for x amount of time? Small battery of course hopefully cost efficient and no larger than the size of my fist.

You could use a drill battery. For example:
http://www.walmart.com/ip/Ultralast-TOOL-929-Replacement-12V-Dewalt-1500mAh-Power-Tool-Battery/35357463
Would likely work. It's rated at 1.5 Ah, and at 120 watt and 12 volt you'd have a current of 10 amps. That means that this battery would (in theory) last you 9 minutes.

get a solar panel
an extra car battery
a converter and something else not sure what its called