Dungeons & Dragons 5e
(Dungeon Master’s Guide p.97) This subclass is included in the DMG to help DM’s build evil paladins to use as villains; as such, before you decide to use an Oathbreaker paladin you will want to check with your DM to make sure they are OK with that. Oathbreaker paladins have broken their oaths to the deity they served (hence the name) and have turned to evil. These paladins get a suite of evil spells added to the normal paladin options, can control undead, have a hateful aura which helps themselves and nearby fiends and undead while attacking, are resistant to damage from nonmagical weapons, and has a super creepy aura that does damage at 20th level. This is an evil paladin through and through and not one which you will want to play (or probably be allowed to play) in a non-evil campaign.
Oath of the Ancients
(Player’s Handbook p.86) Oath of the Ancients paladins are an ancient order, revering nature and beauty and joy. They can use their channel divinities to either tangle their foes in vines or to turn fey or fiends. They also have an aura which protects all friends within it, they return to 1 hit point after being knocked below that, and they do not age (either naturally or magically).
Oath of Conquest
(Xanathar’s Guide to Everything p.37) Oath of Conquest paladins hate chaos above all else. Their channel divinity options include frightening creatures and increasing their attack roll by 10. They also have an aura which instils fear in those in the paladin’s immediate vicinity, and when creatures hit these paladins they are damaged with a psychic blast. These are some seriously scary paladins.
Oath of the Crown
(Sword Coast Adventurer’s Guide p.132) Oath of the Crown paladins are sworn to uphold the laws which hold society together. These paladins can use their channel divinities to either challenge creatures to battle, in which case they must not move more than 30 feet from the paladin, or to heal all injured creatures of their choice within 30 feet. These paladins also are able to take the damage given to another while the attack is happening, and have advantage on saving throws to avoid being stunned or paralyzed.
Oath of Devotion
(Player’s Handbook p.85) Oath of Devotion paladins are all about justice, virtue, and order, just like the original paladins of myth. They can use their channel divinities to turn their weapon holy or to turn undead and fields. They emanate an aura which prevents nearby allies from being charmed, and the paladins are protected from good and evil. These paladins are the pure of heart.
Oath of Glory
(Tasha’s Cauldron of Everything p.53) The Oath of Devotion paladin is all about the glory. Unlike all the other paladins? Anyhow, these paladins are extra strong, they can help their friends avoid damage while doing extra damage, they help their friends to all move faster, and they heal their friends when the paladin hits hard with their divine smites. Glory it up, baby.
Oath of Redemption
(Xanathar’s Guide to Everything p.38) Oath of Redemption paladins use violence sparingly and believe that all people can be redeemed and that benevolence is a better path than violence. Their channel divinity options are giving themselves a bonus to their Persuasion for 10 minutes and the ability to use a reaction to damage a creature which causes damage to someone else. These paladins also have an aura which allows them to take onto themselves the damage that others in the aura receive, and they heal every turn once they pass below half their hit points.
Oath of Vengeance
(Player’s Handbook p.87) Oath of Vengeance paladins seek to punish someone for a crime which they committed. They are able to use their channel divinities to either frighten an enemy or to gain advantage on attacks against an enemy. Additionally, if these paladins strike a creature with an opportunity attack, they can move up to half their speed as part of that reaction, and they can also use their reaction to attack an enemy back.
Oath of the Watchers
(Tasha’s Cauldron of Everything p.54) Oath of the Watchers paladins specialise in protecting the world against the extraplanar entities of Dungeons & Dragons. These paladins are able to help their comrades buttress themselves against attacks from this sort of incursion and boost their initiatives, and they can also turn aberrations, celestials, elementals, fey, and fiends. They additionally cause damage to creatures who unsuccessfully cause your comrades to roll savings throws for INT, WIS, or CHA. Extraplanar baddies beware.