Science fiction characterized by an emphasis on scientific or technical detail, or on scientific accuracy, or on both detail and accuracy. -- Wikipedia
At its worst, hard SF can consists of thinly-disguised science puzzles or even didactic essays, with wooden characters and feeble plots. At its best, however, it can be as well-written, richly imagined, and moving as any realistic writing. Early hard SF was often accused of taking too little account of society, placing characters just like mid-20th-Century Americans in the far future, when society ought to be presumed to have changed. But mainstream hard SF absorbed the influence of the 'New Wave' movement of the '60s and '70s and the 'Cyberpunk' movement of the '80s, so that in the 'radical' hard SF of the best modern writers technological change effects not only profound changes in society, but in the very concept of what it is to be a person.
There is a degree of flexibility in how far from "real science" a story can stray before it leaves the realm of hard SF. Some authors scrupulously avoid such implausibilities as faster-than-light travel, while others accept such notions (sometimes called "enabling devices," since they allow the story to take place) but focus on realistically depicting the worlds that such a technology might make possible. In this view, a story's scientific "hardness" is less a matter of the absolute accuracy of the science content than of the rigor and consistency with which the various ideas and possibilities are worked out.
For the purposes of SF RPGs, hard SF settings have the advantages that players can extrapolate from their knowledge more freely and certainly, and avoid the cognitive dissonance of having to deal with things that they know to be absurdities. On the other hand, hard SF RPGs can be intimidating to players with modest knowledge of science, especially when playing in groups or with GMs who have markedly more knowledge of the subject.
Examples of hard SF role-playing games and RPG settings: