I will note a few of my favorites.
Barbarian: Its entire flavor is wonderful baked into its first few abilities:
1) Rage: Armor class only means there is a chance to stop the pain, Rage means you can simple TAKE IT!
2) Unarmored Defense: I literally stop weapons with my muscles...now we are talking.
3) Reckless Attack: So I can trade puny defense for even more offense! Uh...duh!
The entire barb class showcased...perfectly flavored, wonderful strong.
Bard: For the first time we have an honest to god bard class that keeps the flavor and is actually competent.
1) Full Spellcasting with Good Spells
2) Bardic Inspiration: Powerful AND Flexible. First, I don't have to play an instrument the whole time to use. Hey man you want inspiration, you got it... go off and enjoy for 10 min. Its a solid bonus, a d6 that keeps scaling. And it helps with skills sure...or with an attack roll, or a saving throw, what you need, when you need it. And the icing on the cake, you can see the die roll before using it, so you can decide to use the ability with a decent guarantee that you are using it right when you most needed it. This ability is very well designed.
3) Expertise: Other than rogues, there will be two skills in the game you are the best at. That is easy for DMs to show off.
Fighter:
1) Action Surge: Literally the heroic moment basically baked into the class. People always talk about the DPR potential of this, but honestly in the hands of a creative player, this ability is so much more than mere offense. The chance to save a damsel in distress, to rush over and shield a friend about to die, the possibilities are endless.
2) Second Wind: Again, another "hero's moment" ability, that surge of endurance to do that last act before falling down.
3) Battle Master: I really like the design of the battle master. I think WOTC finally nailed make a fighter that had just enough "pizazz" to appeal to more Book of 9 Swords people but still feels mundane enough in flavor to cater to the "I don't want a spellcasting fighter" people. Its a very hard balance to find, and I think they did well with this subclass.
Paladin:
I honestly think the best changes to the Paladin weren't in favor per say, but they found a way to make the paladin personally powerful but in a more "team spirit" aspect. Also improving the flexibility of the class by removing the LG requirement and the focus on smiting evil.
1) Smite: A powerful weapon that is the literally "cool and special" moment for the class, as everyone is wowed by his spike of damage. Further, but not requiring it on evil creatures, frees up the DM to use other monster types.
2) Aura of Protection: The traditional Paladin trademark, the literally "I am too cool to be hurt". However, now he can share it with his party, which is both very flavorful but also creates interesting tactical decisions (should we bunch up around the paladin for protection but risk a fireball?
Rogue:
Another class WOTC just "nailed" in the first few class abilities.
1) Sneak Attack: What revolutionary about SA in 5e is how simple it is. No more weird situations, having to worry about monster types or concealment or distance or XYZ. Rogue get advantage, rogue get damage.... it just works!
2) Cunning Action: This ability alone encapsulates a million little rogues things in one simple class feature. I am super fast, super stealthy, I can get out of danger. Again such a simple decision that is cinematically so powerful.
3) Expertise: Like the bard, the true master of a few skills.