5E class options

Barbarian: The hitty guy, this class is based around giving and taking damage: You hit more but get hit more, you do more damage and take less. The PHB has Berserker for making more attacks, and Totem Warrior which has more spiritual abilities.
Other books: Sword Coast has a dwarf-only version and more Totems. Xanathar has a different 'Guarded-by-Spirits' option, a storm barbarian who has rage-wind around them, and a Zealot for religious rage.

Bard: As well as full spellcasting, Bards get a lot of skills, skill improvements and general support abilities. The PHB has College of Lore for even more skills and magic, or College of Valor to become a decent warrior.
Xanathar has options for a different type of fighty bard, a bard based more about enchantment and illusion, and a spy bard.

Cleric:Very simple, spells and a Channel Divinity ability for turning undead. The main thing is that your chosen Domain has a bigger impact than previous editions. There's too many options to list, but they all give more Channel Divinity options and other interesting stuff.

Druid: Primarily a spellcaster, they have their Wild Shape ability but its pretty tightly controlled, though you can use it quite frequently. Circle of the Land makes you a better spellcaster, Circle of the Moon improves your Wildshape so you can use it better in fights.
Xanathar has a druid who deals with Fey and one who deals with the spirits.

Fighter: The Standard Hitty Person. They get a ridiculous amount of Ability Score Improvements compared to others, meaning they can either max out stats or get Feats.  They also get fighting styles and other things to make them tanky. Champions are simple fighters who get really high Crit numbers, Battle Masters can choose between special manuvers to knock foes around, and Elderitch Knights can cast spells.
Other books: Sword Coast has Purple Dragon Knight, who leads and protects others. Xanathar has Arcane Archer for magic arrows, Cavalier for horse tricks, and Samurai for courtly manners and charging in heedless.

Monk: Aside from the usual punching powers, Monks get a laundry list of minor but interesting powers as they level up. Some of them rely on Ki points, but those recharge quickly enough. Open Hand monks get more fighting options and inner peace, Shadow monks are ninjas, and Four Element monks are reminiscent of a certain 00s cartoon of some popularity.
Other Books: Sword Coast has lifedrain monks and monks filled with holy light. Xanathar adds Jackie Chan shenanigans monks, and monks better with weapons.

Paladins: The holy hitty boys have a general mix of abilities, both normal and supernatural. Oath of Devotion ups your general Holy factor, Oath of the Ancients makes you more druid-y and defensive, Oath of Vengeance makes you angry and killy. The DMG has Oathbreaker if you want to be a naughty person and deal with demons and undead.
Other books: Sword Coast has an oath for following the king and helping your allies, Xanathar has Oath of Conquest for being a tyrant, and Oath of Redemption for being a hippy paladin.

Ranger: A combination of fighting, wilderness skill, stealth and spotting. Hunters gain greater offense and defense, beast masters gain a companion. This is considered a weak class, and there is a Revised Ranger who has updated abilities on the WotD website.
Xanathar give Horizon Walker, able to deal with the planes, Monster Slayers who focus on one foe, and Gloom Stalkers who work well in the dark.

Rogue: The sneaky types have a lot of defense abilities and are really good with their skills.Thieves are best at sneaky stuff, Assassins get surprise attacks and impersonation, and Arcane Tricksters gain spells and magical tricks.
Xanathar has Inquisitives who figure things out quickly, Masterminds who can get through social situations, Scouts who move quickly out of danger, and Swashbucklers who have sword tricks.

Sorcerors: Have spellcasting and natural ways to alter their spells. Draconics gain dragon features, Wild Magic lets them manipulate dice but means random rolls on a chart.
Xanathar has Divine Soul to gain Divine Magic, Shadow Magic for darkness tricks, and Storm Sorcery for elemental air magic.

Warlock: Freaky spellcasters, with less spells but can cast them far more often. They have a Patron: Archfey, Fiends, or Great Old Ones.
Sword Coast adds The Undying. Xanathar adds The Celestial as a Patron, and the Hexblade option.

Wizards: The basic spell types. Have specialisations for all the spell schools
Sword Coast adds Bladesinging for Elves. Xanathar adds War Magic.