A right, for Hobbes, is a liberty — the absence of obligation. The right of nature is the liberty each person has to use their own power however they judge best for their own preservation. It has no limits: in the state of nature, everything is permitted that serves self-preservation, and nothing is forbidden. This total liberty is precisely what makes the state of nature so dangerous.
A law, by contrast, is not a liberty but an obligation — it determines and binds. The law of nature is a precept of reason that forbids what is self-destructive and prescribes what best preserves life. Right and law therefore run in opposite directions: where right says you may do anything, law says there are things you must do and things you must not do. Hobbes insists this distinction, routinely blurred by earlier writers, must be kept sharp.
Reason, observing the catastrophic consequences of the state of nature, delivers a clear verdict: seek peace wherever it is attainable. This is the first and fundamental law of nature. But reason is also realistic: when peace is not attainable, it permits — indeed requires — that you use every advantage of war for your self-defence. The law of nature is not pacifism; it is a rational instruction to prefer cooperation when cooperation is possible and self-defence when it is not.
Hobbes derives a sequence of further laws of nature from the first — be willing to lay down the right to all things when others are willing to do the same; keep covenants; be grateful; accommodate oneself to others — arriving eventually at the summary law that all the laws of nature amount to one rule: do not do to another what you would not want done to yourself. He calls the laws of nature not moral commands but theorems of reason — conclusions that any rational agent would reach by reflecting on the consequences of their actions. Their force is not the force of divine command but of rational self-interest properly understood.
The distinction between right and law appears at the opening of Chapter XIV of Part I. Hobbes derives nineteen laws of nature in Chapters XIV and XV, culminating in the summary rule that anticipates the Golden Rule in a decidedly un-theological register.