"Forgeite" refers to users and game designers who frequented and followed design theories of a site known as "The Forge," found at http://www.indie-rpgs.com [Internet Archive link]. While the site is now defunct, it had a profound effect on game design and theory among independent game designers, with two of the the most notable being D. Vincent Baker (Dogs in the Vineyard, Apocalypse World) and Ron Edwards (Sorcerer).
Ron Edwards is also known for the GNS model of role-playing games, which suggests players fit into one of three categories based on which approach to role-playing they take: Gamist, Narrativist, and Simulationist. The theory discusses game design with these approaches in mind. I am not really qualified to go into detail on this model, but it spawned a lot of debate and heavily influenced many games which came out of the Forge and many Forgeite designers.
The most profound effect was on game design itself. One of the hallmarks of games from the Forge and its designers is the highly focused mechanics designed to promote and to fit within the premise of the game. This isn't true of all indie or Forge-inspired games, but this design theory found its way into many games of the time and current games.
The OP mentions Dogs in the Vineyard, a game about wielding authority to serve the greater good. The mechanics of this game are designed with the goal of pushing conflict into ever more risky levels, encouraging players to consider the increased risk at every turn. You start in discussion, and can escalate through several levels, until you have to decide if a matter is serious enough to warrant use of deadly force, with all its attendants risks. The game is all about "how far will you go?" and the rules are designed to force that question as often as possible.
Another effect, which is only my opinion based on observing the market, is that The Forge helped create the wide open RPG market we have today by fostering and encouraging many designers to put their works out into the world. The advent of electronic publishing helped here too, but it takes more than just an easy avenue of distribution. It takes a lot of encouragement to get many to take that first step, and the Forge had a hand in that.
Regarding the emotional component, I don't really see that as a common primary goal among Forge and Forge-inspired games. Some games seem designed to explore certain emotions, but I see such as typical of the specific focus; in these games the narrow focus, supported by specific rules, is the exploration of emotion.
For example, Emily Care Boss wrote games which explored various aspects of romance. The rules (and fair warning, I've read two but never played any of these), appear designed to explore those emotions associated with romance on various levels: "Breaking the Ice" looks at new romance; "Under My Skin" explores secret feelings, hidden passion, etc. The emotion is the focus in these cases, but the narrow focus is what marks it as a Forge-inspired game.