Each of the Homestuck Kids and the Trolls has a mythological role, as hinted at during various points of the Homestuck story. All titles appear to consist of two components - a Class and an Aspect. A title can be assembled from any combination of class and aspect - John in the kids' session is the Heir (Class) of Breath (Aspect), while in the Trolls' session Tavros is the Page (Class) of Breath (Aspect) and Equius is the Heir (Class) of Void (Aspect).
Aspects are a form of generalized "Elements", in that they are primal forces of reality. Classes describe mechanics and intended roles, "channeling" the Aspect into possible action and powers. Classes come in active(-)/passive(+) pairs, with the active employing or affecting his aspect directly and for his own gain, and the passive being affected by his aspect, and sharing his actions with the team.
Do note though, that literal interpretation of the aspect's/class' names is misleading, i.e. "Prince" actually means "Destroyer", and "Light" means "Fortune".
Heroes with different aspects can end up with very similar powers depending on their class. The Life and Doom aspects may be radically different, but depending on the class their powers may end up being similar.
Player abilities may also manifest in ways contrary to their aspect if they are heavily resistant to their true calling, or if an outside influence corrupts them. This happened to Rose Lalonde, and may have also occurred to other players in the Troll Session (as their entire society was corrupted by Doc Scratch and Lrd English).
The term "Hero" can be used in lieu of the class when the speaker wishes to be generic, i.e.: John and Tavros are "Heroes of Breath". Both components of the title appear to have some mechanical effect on the game, depending on the class or element assigned. The Salamanders of the Land of Crypts and Helium ominously use the term "Noble" to refer to their Heroes, as Nobles suffer and become martyrs, instead of the way Heroes save people.
Sburb may design planets with the titles in mind. For the kids one part seems to come from their aspect, while the other comes from their Associated Classical Element, or a force actively opposing it (Dark clouds in John's case and Frost in Jade's case). The Troll's lands mostly relate to their interests, but some show a clear connection to their titles.
Doc Scratch implies that every session will have a Hero of Time, and if the breeding duties are indeed linked to the Hero of Space they would be a constant in the game. Calliope has confirmed that these two aspects are indeed the minimum requirement for every session.
On the scale of classes, there are at least two "rare classes" referred to as Master Classes. One such master class, the Muse, is one of the only two known classes, and even then there is no other relevant information other than that it is "the most passive class" and strictly for females. Its counterpart, Lord, is the male master class which has been stated to be the most active class.
It is entirely possible that the number of players in a given session will determine what titles the game can and will assign. It appears games with 4 or more eliminates the chance to receive master classes, which may serve the purpose of fulfilling multiple roles in under-populated games.
Prophetic visions involving utter ruin, failure, and death. (? theorized based on his screen name, his multiple deaths, and the similar fate of The Ψiioniic)
Doomed to die many times, and has overcome death on more than one occasion (thanks to double dreamselves and "half-ghost" form)
In one of Karkat's memos, Kanaya speculates that Skaia assigns titles to players to challenge them, rather than to suit their strengths.
This would certainly explain some of the apparent inversions, such as Rose being the Seer of Light despite her fascination with darkness and horror, and Jade's title as the Witch of Space despite her life never leaving her small island. However, her theory should be taken with a grain of salt, considering the titles of a lot of the players match them perfectly.
It could be speculated that players are meant to "grow into" their roles; that is, when they first begin the session their title and its duties force them outside their comfort zone and seem very unsuited, but a fully realized Hero has matured into the role the game has assigned them.
Trivia
According to Andrew Hussie, other characters have also assumed these titles, or otherwise will when they ascend to the God Tiers.
When John, Jade, and everyone on the meteor arrive in the Alpha Kids' session, their shared session will have one Hero for every listed aspect but Space (of which there are two: Jade and Kanaya) and Doom (of which there are none, since Sollux left with Aradia). This session will have all classes as well, except for Mage (again due to Sollux's departure) and technically Thief, but Tavrisprite is an unknown entity at this point.