First of all what are gems? Gems are little jewels that have magic power, some of them are weaker and some stronger. They can increase an attributes from your equipment.
Where do I get those? Gems are found by mining trough ores like coal. You will notice that there are three different types of gems to be found yet. The only difference between those is the cut. Here are pictures of these, you can tell the cut not only by looking in the lore. You can see it at the item and its colour.
How do I use them? When you hover over your amour and it has a gem slot, you need to check the cut that is needed for that. If they and the level of the gem and amour match, you can put the gem on by just simply left clicking it with the gem on it.