A mod, or an APV (advanced personal vaporizer), is a regulated or unregulated step-up battery from an Ego style battery.
Typically, a regulated mod has a replaceable battery that needs to be taken out and plugged into a charger to recharge before being reinserted. It also contains a few electronics and wires to prevent damage to the battery or to the user. All mods are typically much larger to accommodate such a battery, but of course have plenty of customization to swap out for a shorter battery. Mods come in tube shapes, like a long fat cigar, and also box-mods, which are basically about the size of a small pack of cigarettes almost. The more you look into it, the more interestingly shaped mods you will find.
Some mods will have features such as VV (Variable Voltage) which will allow you to turn up or down the voltage in order to supply wattage to your tank more evenly. This is quickly becoming the standard in all mods lately, as tanks are becoming more and more diverse.
A step beyond a regulated mod is an unregulated, or better known as a "mechanical mod". These are basically hollow metal tubes made out of stainless steel or brass that have no wires or built in safety features to protect the battery. These are for experienced vapers only, as you need to know exactly what you're doing to operate one correctly.
To answer your last question, a T3S clearomizer has Ego threading, this is why it fits on your Ego battery. Most mods you see today will have Ego AND 510 threading. This means it can fit a clearomizer like the T3S on top, or even a 510 tank such as the Kanger Protank 2 for example. Before you order a mod, be sure to check what threading it has.
Hope that answered most of your questions!