There are, broadly speaking, two kinds of damage dealers. There are the ranged damage dealers and the melee damage dealers. As a whole, damage dealers do just what the name says — they deal massive amounts of damage to their enemies without drawing a lot of threat (also called “hate” or “aggro”). Ranged damage dealers generally consist of casters or classes that use ranged weapons such as guns or bows. Melee damage dealers generally need to be standing in close proximity to the enemy in order to hit it since they are using knives, their fists, swords, or other short-range close-combat weapons.
Damage dealers do not take a lot of damage — though they generally have a bit more survivability than a healer, they are not tanks with enormous health pools and a lot of shields or mitigating abilities. And, while some damage dealers do have healing spells or the ability to raise fallen allies, they are not as proficient at it as healers. What they are best at is cranking out insane amounts of damage, climbing the damage meters, and helping to kill the enemy fast before the healer runs out of MP and the tank runs out of HP. So, if big numbers, optimization, min/maxing, and killing things fast appeals to you, then this may be the role you like best.
Damage dealing classes in FFXIV: Melee: Pugilist (becomes Monk), Rogue (becomes Ninja), Lancer, (becomes Dragoon), Samurai (requires Stormblood) Ranged Physical: Archer (becomes Bard), Machinist (requires Heavensward), Dancer (requires Shadowbringers) Ranged Magical: Thamaturge (becomes Black Mage), Arcanist (becomes Summoner), Red Mage (requires Stormblood), Blue Mage (limited job, requires Stormblood)