MG stands for milligram and is the measure of the concentration of nicotine in the vape juice. The beauty of vaping is that you're able to choose vape juices with different nicotine levels (compared to cigarettes, which usually have a standard amount).
If you see 3mg on your vape juice bottle, it means that the juice contains 3mg of nicotine in every ml ie. a 50ml bottle of vape juice would have 150mg of nicotine.
Vape juices usually have nicotine levels that range from 0 - 24mg. Heavy smokers might start at the highest level and gradually reduce the concentration so that they can eventually quit nicotine altogether.