Welcome Shaun,
It's a huge huge topic, but I'll try and answer as succinctly as possible.
With modified e-cigs, you will come across a variety of batteries, most take one or two different types, the GGTS, takes them ALL.
Batteries vary greatly in size and capacity.
The first two digits refer to an approximate diameter and the third and fourth two refer to an approximate length, the last 0 denoting that the battery is a round shape.
For example, 18650 batteries are approximately 18mm in diameter and approximately 65mm in length - although these dimensions vary from battery to battery and also between protected and unprotected versions.
The larger the battery is, the more capacity is usually has, the capacity is expressed in mAh. Average users can get a day or two of vaping out of an 18650. From your other forum posts, it sounds like you would not want a battery as big as the 18650, or the 14500, so for your purposes, you're looking at a 10440 battery on the GG slim. These are usually sold at around 500mAh but in practise are probably nearer 400mAh (or maybe even less). Capacity is often mis-advertised on Chinese batteries and there's not a lot we can do about it unfortunately.
So, comparing a 10440 to a standard 808 e-cig battery for example, you'd get about 2.5 times more vaping time on a 10440 than on a standard 808 battery.
What's more, standard li-ion batteries tend to be cheaper than Chinese e-cig batteries and you can pick them up at many more outlets including flashlight shops etc.
Hope this helps with a few of your questions.