The Souls series in interesting in that it has had influence on gaming culture as well as actual games.
When Demon’s Souls came out in 2009, games were very much designed around the whole “make you feel awesome/powerful” and “dying is but a minor obstacle with no longer-term consequences” principles. DS challenged that whole mindset. Whether it was an entirely intentional challenge remains in question, because it was kind of just following some older design tenets with a more modern game design (being a “spiritual sequel” to King’s Field etc.). (1)
So that started a conversation, and Dark Souls popularized that conversation, about what people actually want from games. There was some stupid initial “I’m cool because I play HARD games” bullshit, but that actually died the death pretty rapidly, and we moved past that to people being able to say that they honestly liked more or less punishing games, and the success of the Souls series allowed designers to think “Maybe it’s okay to make the game pretty tough” in a way that had previously been seen as unacceptable.
That conversation is still going on, of course, but the resurgence of Rogue-loose games and “losing is fun” games (which post-dates Demon’s Souls, for the most part - DF was around before, for example, but wasn’t big until after), has, together with the Souls games, helped to open up ideas of acceptable or fun game design in a way that I think is leading to more people being happier and game design being more diverse.
There are also specific game elements being copied - like the concept of multiplayer where enemies just appear in your SP levels/areas, or lore that is in revealed almost entirely by in-game stuff, not cutscenes or exposition, as noted by Wenz. Wenz also has a good list of games more directly inspired/influenced by the Souls games. There are some others, too.
This article may also be of interest: http://kotaku.com/what-game-designers-love-and-don-t-love-about-souls-g-1699490162
(1) = For me personally, Demon’s Souls was one of the most refreshing games in a very long time. Not because I’m some awesome hard bastard. I’m like, better than average at games, but only because I play them an awful lot. Rather the whole willingness of the game to metaphorically “kick me in the face”, in a way that was exciting rather than merely grindy/tedious (unlike meaner older MMOs, for example), was awesome. I was actually on-edge in a way that I hadn’t been for a long time.