Claim rights and liberty rights
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A claim right is a right which entails responsibilities, duties, or obligations for other parties. By contrast, a liberty right is a right which does not entail obligations for other parties.
A liberty right to x is permission for one to do or have x, while a claim right to x is an obligation on others to allow, enable or provide x. For example, to assert a liberty right to free speech is to assert that you have permission to speak freely; that is, that you are not doing anything wrong by speaking freely. But that liberty right does not in itself entail that others are obligated to help you communicate the things you wish to say, or even that they would be wrong in preventing you from speaking freely. To say these things would be to assert a claim right to free speech; to assert that others are obliged to refrain (i.e. prohibited) from preventing you from speaking freely (that is, that it would be wrong for them to do so) or even perhaps obliged to aid your efforts at communication (that is, it would be wrong for them to refuse such aid). Conversely, such claim rights do not entail liberty rights; e.g. laws prohibiting vigilante justice (establishing a legal claim right to be free thereof) do not thereby condone or permit all the acts which such violent enforcement might otherwise have prevented.
However, a liberty right can be asserted as the inverse of a claim right: a person has a liberty right permitting him to do something only if there is no other person who has a claim right forbidding him from doing so. This is because the deontic concepts of obligation and permission are De Morgan dual; you are permitted to do all and only things you are not obliged to refrain from.
Another oft-cited example is picking up a shell at a beach. If a person has a liberty right to pick up the shell, he is morally permitted to pick up the shell; but no one is obliged to help him do so, or even to refrain from preventing him from doing so. The person would be doing nothing wrong by picking it up, but no one else would be doing anything wrong by preventing them, either through action or inaction, from doing so. By contrast, if that person had a claim right to pick up the shell, others would be morally forbidden (obliged to refrain) from preventing him from picking up the shell; or, if that person had a claim right to have such a shell, others would be morally obliged to provide him with such a shell.
A world with only liberty rights, without any claim rights, would by definition be a world wherein everything was permitted and no act or omission was prohibited; a world wherein none could rightly claim that they had been wronged or neglected. Conversely, a world with only claim rights and no liberty rights would be a world wherein nothing was merely permitted, but all acts were either obligatory or prohibited. The assertion that people have a claim right to liberty - i.e that people are obliged only to refrain from preventing each other from doing things which are permissible, their liberty rights limited only by the obligation to respect others' liberty - is the central thesis of liberal theories of justice.
The distinction between liberty rights and claim rights should not be confused for the distinction between negative and positive rights. Both liberty and claim rights come in positive and negative varieties: your permission to do something is a positive liberty right, your permission to refrain from something is a negative liberty right, another's obligation to do something for you is a positive claim right, and another's obligation to refrain from doing something to you is a negative claim right. However, the De Morgan dual relationship between claim rights and liberty rights crosses the positive-negative rights distinction: one's positive claim rights limit others' negative liberty rights and vice versa (i.e. others are obliged to do something for you if and only if they are not permitted to refrain from doing so); likewise, one's negative claim rights limit others' positive liberty rights and vice versa (i.e. others are obliged to refrain from doing something to you if and only if they are not permitted to do so).