I have heard usage of the word "Forgeite" as a descriptive name for a certain kind of games - in most recent example it was Dogs in the Vineyard. It is implied that it has something to do with Forge and that games like that have an agenda of creating certain feelings in the player - but I'm not sure if that's correct.
What does the term "Forgeite" mean? Why is it significant? Where does its usage come from?
"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.
The Forge was an RPG discussion forum with an emphasis on developing theories and terminology to describe RPGs in ways that allowed for both academic criticism, and practical advice to gamers and especially indie game designers. Setting aside some contentious personalities and statements, their main contribution to the RPG landscape seems to have been fourfold:
A model for analyzing issues of group dynamics and other problems that crop up in RPGs. Fleshing out the concepts above naturally involved acknowledging that people play RPGs for different reasons and with different intents, which led to the idea that most problems and bad experiences with RPGs stem from a lack of agreement about what game is actually being played and what experience the players and GM are trying to create, aggravated by an emphasis on maintaining mystique and "immersion" at the expense of frank discussion. This consensus-based model for analyzing social issues turned out to be pretty useful, and led (on a later Forgeite blog) to the Same Page Tool, much beloved by some of us here at RPG.SE as a way to prevent (or, in a pinch, resolve) table arguments during play by agreeing in advance on what a particular game and group are going for in terms of play style, fictional style, etc. Often just pointing out that there are different styles of play, all equally valid, is a real eye-opener and helps people work out their differences. (This model also leads to a tendency to advise that groups who aren't having fun together shouldn't play together, the appropriateness of which varies.)
A focus on specialization as an attempt to avoid such problems, based on the belief that the lack of agreement outlined above kept happening partly because mainstream RPGs, as designed and written, failed to provide sufficient clarity on several questions necessary for satisfying gameplay. The conclusion of this line of thinking, which is not universally accepted in the broader RPG community, went so far as to posit that even if a given set of players could theoretically enjoy different styles, a given game can only succeed by focusing on encouraging one type of play, with the alternative being "20 minutes of fun in 4 hours" as broader games attempt to alternate between the experiences that different players want but inevitably fail due to lack of coherence as different mechanics contradict each other.
So rather than having everyone get some of what they want, typical Forgeite games are narrowly focused on giving a certain type of player exactly what they want, under the theory that players who want something else can be elsewhere playing a different game that gives them exactly what they want. Even if those players are in fact the same people at different times or in different moods, the idea is that it's best if all the instructions and mechanics are tuned toward creating a particular experience so everyone knows exactly what they're signing up for. (Focus this tight tends to also lead to shorter games designed for one or a few sessions, not campaigns of many years; the variety comes from playing different games, not trying to create different experiences within the same game.)
The opposite philosophy, as implied by advice in the D&D 5E DMG and by the existence of "universal" RPGs, is that it is possible to give everyone what they want at once, even if they want different things, for example by having compelling character- or story-based reasons for getting into combat. This is sometimes called the "Big Tent" philosophy. In between, and probably more realistic than either, you have the idea that plenty of players enjoy several different aspects of play enough to make the game enjoyable overall, even if not every moment can be their favorite, and that this is sometimes necessary given the limited number of RPG players and GMs in most areas.
A number of indie RPGs, many of which took these ideas to heart and are highly focused in terms of play experience, setting, and genre. Less definitionally but as a broad pattern, a lot of them specifically focus on the Narrativist angle and player-driven storytelling, which had arguably been neglected. Either way, they tend to specify, or provide explicit tools for specifying, parts of the game experience and social contract that more mainstream games had left unsaid. Some of these have done pretty well, and Dogs in the Vineyard is a popular example. This author also argues (right before "What are the alternatives...") that this concept of specialization is also visible in D&D 4E, and even as 5th edition returns to being a broader game, there's a bit more explicit discussion of player types and motivations than there had been, suggesting that the influence of Forge theory isn't necessarily limited to the indie games created directly by forum participants.
There are two great answers already, but I'm coming from a different perspective, based maybe on my understanding of the Italian RPG scene and on having passed a significant share of my free time at the Italian counterpart of The Forge, a forum called Gente che Gioca (lit. "People who Plays").
In our panorama, a forgeite game is often classified as such by people who don't like them, in an attempt to gather them all under the same hood in order to diss them as a whole.
More often than not, these people were offended by attenders of the Forge who told them that all the most popular games (D&D, GURPS, WoD, Call of Cthulhu) are basically the same game because (as written) it plays the same (you ask the DM if you can do something, the DM tells you if you can, if you can not or what to roll in order to know, then the DM decides the outcomes anyway); people who got offended by Ron Edwards stating that all those who played those "dysfunctional games" have been dealt "brain damage" because of the unhealthy power relationships between GM and players (mostly, since the GM determines the outcome anyway, to gain authority on the fictional world you need to apply social pressure on him); or people who were not OK with categorizing the focused games emerging from The Forge as RPGs because in RPGs you can do whatever you want.
At the same time, people who like the games generated by the Forge often wear the name proudly, turning it into a distinctive flag, but they don't usually classify games as such because they strongly believe that every game is a different story and categorizing them under a single label does more harm than good.
tl;dr: as a game category, the name "Forgeite" arguably comes from detractors trying to fit the games generated from authors involved at The Forge, or inspired by the Forgeite movement, under a single label.
