There are two main kinds of rights being asserted in the debate over fracking. These are property rights and, for want of a better term, human rights.
On the one hand, there is the side of the debate that believes that they should be allowed to conduct fracking operations. This side tends to assert property rights. They argue that people in a democratic and capitalist system have the right to do as they wish with their property. They argue that a person who owns a piece of land has the right to conduct whatever activities they want on (or under) that land.
On the other hand, there is the side of the debate that opposes fracking. From this point of view, the issue is one of human rights. Specifically, people on this side of the debate would argue that they have a basic human right to have their health and safety protected. They argue that fracking would endanger these rights by causing pollution or even earthquakes.
Of course, these are not the only arguments being made on each side. Much of the argument over fracking is over the effects of fracking, not over rights. However, these are the general types of rights that are being asserted as part of this argument.