not grams. units of g is acceleration of gravity (9.8m/s^2)
http://www.teamliquid.net/forum/tech-support/333648-an-overview-of-mouse-technology
basically with acceleration when you move your mouse fast, it moves more distance per inch of your mouse movement while when you move the mouse slow it travels less distance. having no acceleration would be a static value of move X distance on screen per distance of X mouse movement. why have acceleration? it allows fine control when you need to while quick movement when you dont want to wait. its a good concept but does not always play nice for gamers who want complete control over their mouse.
most mice have a small amount built in but windows has an option to turn off at least the software portion of it (windows adds extra). different mice have different levels of acceleration as well.
no, your mouse will not go all over the place, it will go in the direction you point it although it will move faster and go further the faster you move the mouse if acceleration is on. as far as how quick and how far that depends on A) your dpi settings B) your acceleration of the mouse C) your windows acceleration settings and D) your mouse speed and game mouse speed/sensitivity options.
keep in mind that certain high contrast mousepads like white text on black background may cause mice to track irregularly but thats a different thing altogether.
i personally use a deathadder 3.5g (old model.. i like it because it is rubberized). while the sensor isnt the best you can get it works for me fine enough and is comfortable. i've tried the proteus and its also a good mouse but was not quite wide enough for my hand and i did not prefer the mouse wheel. otherwise it was quite excellent and tracked well with easy to locate buttons. its hard to go wrong with the other logitech gaming mice as they have been a staple of gaming for many years. roccat kone is worth looking at. not sure of the acceleration on all of them, i'll leave it up to you if you are worried about it. quite frankly... i've never bothered and have been fine