There are some pretty unique terminology used in FFXIV, which may make some players puzzled. And here, I'm going to decipher them. Mainly about why they're important and what they do for you.
A FATE stands for Full Active Time Events. Basically, they're public quests that occur randomly across Eorzea; much akin to Guild Wars 2's dynamic events. Just by opening your map, you'll see any that are currently active, there intended level, and remaining duration. There are many different FATEs - some require you to kill waves of enemies, defeat one giant boss - like creature, defend an NPC and collect items.
Depending on the amount of input in a FATE, you'll be awarded with either a bronze, silver, or gold medal, with scaling Gil and experience. You can also participate in a FATE even if you're over the recommended level; if you're more than five levels ahead, you'll be required to sync your level down to be able to reap any sort of reward.
Guildhests are nothing to be afraid of. When you're level 10, you'll unlock your first one, which can be accessed via the Duty Finder. They're similar to dungeons, in that they require a party to complete, but often just focus on one objective. Apart from a big one - time bonus for completing Guildhests for the first time, they're primarily intended to teach new MMO players on working together, and about their roles of tanking, DPS and healing. Regardless, they're worth doing for new players and veterans alike.
Finally we come to Guildleves - they're just different name for repeatable quests. There are different types of these quests: Battlecraft Leves are quests involving dispatching enemies, and have the option to have their difficulty increased for better rewards. Tradecraft Leves require you to hand in specific crafted items, and if the player hands in high quality items the experience reward is doubled. Fieldcraft Leves give players the task of gathering materials within a certain time limit. You only have a limited allowance of Levequests - players gain three allowances every 12 hours, stacking up to 100.