A  positive right is a right *to* something, a claim. This may be a social (right to education) or political (right to vote) right. A negative right is a right against interference, a defense. An example is the first amendment's prohibition to restrict free speech. 

In general, and regardless if negative or positive, rights can conflict with each other and should be seen as a system that has to be brought into balance or even into an optimum. Moreover, a negative right can usually be expressed as a positive right (right to free expression as prohibition to restrict free speech), but not vice versa.