Note: Every enemy except for the Shank has a variant for the different "species" in Destiny 2.
The first species is the Hive. Other than the original versions of these enemies, the first species you encounter is the Hive. The Hive enemies look almost paper-like in appearance and the only difference between them and the normal enemies is the health and damage they deal.
The Taken enemies appear in The Pale Heart, their appearance like a super bright star for their face and a galaxy for their body. Being the last enemy species in the first chapter they have the most amount of health and damage out of all of the enemies in the first chapter.
The Vex are in the second chapter as the first new species in that chapter looking like robots with a single red eye in the center of their head.
The Fallen are very very similar to the normal enemies only slightly stronger.
Unstoppable Champions are champions that never stagger (unless a certain type of subclass is used on them), which is when they get hit for enough damage that they stop moving for a short time.
Barrier champions are champions that have a stronger shield. Every champion has a sheild, which is basically an extra bar of health that they can regenerate after a short time.
Unstoppable champions are extremely aggresive, chasing to the other end of the map to hunt you down. Along with that, they have a 70% damage reduction property. This means that they take 70% less damage from all incoming damage sources. The only way to remove this property is to stagger them, whereupon the damage reduction is removed.