Each class in Destiny 2 has three subclasses, unlocked as you progress. For clarity I will split them into three groups:
1) Destiny 2 subclasses;
Hunter - Arcstrider
Warlock - Dawnblade
Titan - Sentinel
These subclasses are brand new for Destiny 2, and are unlocked very early in the game in an unskippable story event.
2) Vanilla Destiny subclasses;
Hunter - Gunslinger
Warlock - Voidwalker
Titan - Striker
These subclasses are from year 1 of the original destiny, but have been modified. To unlock these subclasses in Destiny 2 you must first reach level 7. At this point you will have a chance to find artifacts in chests in patrol. These can be gold regional chests, or the reward chest for completing a public event or HVT encounter.
Upon acquiring this artifact, you will revieve a new milestone instructing you to charge your power by defeating enemies with fellow guardians in public events, however I noticed the percentage went up from kills in any activity, even when solo.
After reaching 100% on the milestone, return to where you obtained your first subclass for a short, simple mission which I will not spoil here. You now have a second subclass.
3) Taken King subclasses;
Hunter - Nightstalker
Warlock - Stormcaller
Titan - Sunbreaker
These subclasses were introduced in Destiny with the Taken King expansion pack. They are earned by the exact same method as the second subclass, but you must be level 15 for the artifact to drop in patrol. EDIT: I never got these to drop before level 15, but several people have reported getting them at level 14.
[Note]: I am a warlock, and have been playing with a titan and a hunter. We all had exactly the same requirements for unlocking the subclasses, so there is no difference based on your class. These are however our first characters, so it COULD be slightly different for your second or third. If so, simply follow the instructions in the milestone you are given upon collecting your artifact from a chest.