Dark Knights belong to the third tier of jobs, presented by the Water Crystal. The concept of a Dark Knight was introduced in Final Fantasy II with Leonhart. However, III marked the first appearance proper of the job.
Well, kind of.
Dark Knights are most proficient at using Dark Blades, a type of blade said to be forged from the very darkness itself. In their original portrayal, Dark Knights could only use those blades and similarly "cursed" equipment and the only other jobs that could use dark blades, in turn, were Ninja and Onion Knights. Supplementry material suggests that they are an opposite to the Knight job: Whereas Knights use the positive aspect of the sword, Dark Knights use the negative aspects. Despite this, they can still use White Magic, albeit only level 1.
Dark Knight was heavily reworked in the remake to more closely resemble their future iterations. They lost their ability to use White Magic in favor of the Souleater skill: A command that strikes all enemies at the cost of a Dark Knight's health. They were also granted the ability to use various other swords and equipment, although Dark Blades were still #1.
On the surface, Dark Knights are native to the village of Falgabard, a village hidden from the rest of the world by a mountain range; In fact, the majority of the NPCs in the village are Dark Knights. Lore in the Famicom NTT Publishing guides give a bit of backstory to the village, revealing that it came to be when the Dark Knight Leonhart was banished from Saronia by the king at the time. He settled in the mountains and was slowly joined by more Dark Knights, until they ended up forming a village. Meanwhile, in the 3D remake, Saronia is said to have searched for the village to learn its secrets and Ingus is heavily implied to be from the village in unused text.
In the 3D remake, Souleater is similar to its appearances in other FF games: The Dark Knight strikes all enemies while taking self-damage of around 10% or so. The damage is respectable enough to be useful in fighting full enemy parties (and is a boon in the Cave of Shadows), but isn't as useful as it otherwise would be in other versions due to the "three enemies per fight" thing. However, a harder-to-notice benefit is that Souleater ignores enemy defense, relying largely on the weapon's attack value, the Dark Knight's job level and the Dark Knight's current HP.
The Pixel Remaster replaces it with Bladeblitz. Based on the recurring skill of the same name, this means that the skill had its self-harm effect removed. As a result, there's little downside to spamming it in battles with multiple enemies.
Bards are a Dark Kngiht's best friend. One of the bard's songs can heal Dark Knights roughly the same amount that Souleater does. Since the bard always goes first, healing up from a Souleater is trivial. Additionally, Souleater is technically speaking not a physical attack. This means that it properly scales with ordinary swords without losing effectiveness like other swords normally do as used with a Dark Knight. This also means that it can be used while mini with good results.