Throughout the course of the original Destiny, players unlocked subclasses that would add an extra element of customizability to their chosen class. With more being added over time as the game matured with add-ons and expansions, you're going to have to fight to earn them back in Destiny 2.
Similar to other MMOs, Destiny 2 employs a system that locks more powerful abilities behind a level gate. Whether you've transferred a character over or not, you're starting fresh. So you Sunbreaker or Stormcaller abilities from the first game aren't immediately available.
For those starting fresh, the first thing to do when making your Destiny 2 character is to pick a class; Hunter, Titan, or Warlock. Starting in the Arcstrider, Sentinel or Dawnblade subclass/spec respectively, you'll only be able to utilize the skills of other subclass/specs as you level up your character. By level 7, you should start to see Class Relics drop from chests on occasion. With the first relating to your second subclass, the next reported to drop at level 14/15 will award you with your third, and final, subclass. But you still have to work for it.
Upon picking up one of these Class Relics, it's time to charge the damn thing. Though the Milestone menu suggests charging the item through public events, players have noticed it charges through regular play - meaning PvE or PvP is a viable strategy. Though the aforementioned public events are the quickest way to do so, you're not limited to playing the game any one way for a class unlock. Once maxed out, you'll be asked by the Shard of the Traveler to simply kill a set number of enemies with your new-found Super ability. Do that, and you'll be granted permanent access.
And that's all there is to it. Typically, each subclass has a range of new abilities and traits to fundamentally change how any one class plays. If you played the original Destiny, you likely remember Stormcaller Warlocks zapping you to death in PvP, or Sunbreaker Titan's running off to toss hammers in PvE. Expect much the same, only at higher framerates once the PC version launches next month.