Negative rights are the rights that we are most familiar with. They give us the right to be free from certain actions by others. For example, they forbid other people to discriminate against us on the basis of our race. They prohibit the government from banning our religion. These protect us from things that others might want to do to us.
Positive rights give us the right to demand some particular action from government or from another person. The government or person has a duty to act in a certain way. For example, there are some who believe that people have a positive right to a certain, basic standard of living. According to these people, a government has the duty to provide us with what some might call welfare payments; that is, it must give us the means to fulfill our most basic needs.