Quartz is a polymer/resin with "stuff" put in it. Some of it look sort of like rock, if most of the material is stone and small pieces of other natural product, but most of it doesn't. It can look like whatever you want it to depending on the colour of the polymer and the material in it (for example a lot of quartz tops are full of, for want of a better word, "glitter").
It certainly doesn't replace "granite" and other natural stone worktops, but is a good alternative depending on what you want.
We're planning on replacing the worktops in our kitchen and bathrooms in the near future and I can't decide whether to go real stone or quartz. Quartz is much easier to maintain but I don't really like the look of most of it, especially the stuff that's pretending to look like real stone. That's just personal preference though.