Welcome to Eorzea! I hope you're enjoying yourself so far!

Everyone above did a great job describing looting and limit break so I won't go into that. However, the gear you are winning may go in to your "armory chest", an area separate from your main inventory that holds he's, including everything currently equipped. To check if you have gear in there, you can choose the item slot in your character screen and see what is there, or go in via the Armoury Chest option, underneath the inventory option.

I personally prefer my loot drops to go into my inventory where I can quickly see what I got. If you find your gear is in the armory chest instead of the inventory, you can change gear to be deposited into the inventory instead under the Character Settings window.