An "aug" is short for an "augment." It is an item that can be attached/linked to an equipment slot on your character. The augment item has it's own stats (like STR or Hit Points or Mana), sometimes even a focus effect. When linked to the item, it adds its stats to that of the item. Doing so provides an opportunity to improve your characters statistics without necessarily changing its existing equipment.
Each piece of equipable armor has one or more augments slots. If you right click on the armor piece (or weapone) and look near the bottom of the item window, it will show you one or more augment slots (usually it looks like "Slot 1, Type 7"). Each augment slot has a type (type 7 is typical for armor, and type 4 is typical for weapons...I'm guessing you don't need to worry about many others now). The augment also has a type. The augment must go in a slot that matches its type (ie, a type 4 augment in a type 4 slot). Some augments have other restrictions. Right click both the item and the augment to make sure their types match, and there are no restrictions that mismatch. When you try to equip the augment to the item, you'll get a warning message if there are problems.
Augments are equipped by using augment pools (they're often knows as "bird baths" because they tend to look that way). There's one at each LDON camp, one in the bazaar, and one in the Guild Hall. I think there's one at the gypsy camp in Lavastorm, as well. Left click on the bird bath and you'll get a window with two slots in it. Put the item in one slot and the aug in the other and hit the "combine" button.
When you're ready to remove or change an augment, you'll need a special item called an augment solvent, which is usually sold by NPCs near the "bird baths."
Augments drop off of creatures, usually in the GoD expansion and later. Many can be purchases using DoN crystals at the DoN vendors. Slot 4 weapon augments can be purchased for melee characters in most starting cities (do a find for "Melee Augments" for a particle trail to follow). There are some excellent quests that result in augments. Do a search (use the "quest" link on the menu to the left of this post) on quests related to the Prophecy of Ro expansion for some good choices. Many basic augments drop in LDON adventures; it can be hard to find people for LDONs these days, but they're great fun and an excellent source of learning how to play the game with a group. If your character is between level 20 and level 40-45, LDONs can be very good experience.
If many of your items are not currently augmented, the above sources may allow you to substantially improve your character without spending more on armor.
I hope this helps a little. It's not everything you need to know, but it will getyou started. You can probably find a friendly person in-game (a group-mate or a guild-mate) to help you, as well.