Nonmaleficence is a moral law that says how you shouldnt do evil. You shouldnt do actions that cause more evil than they remove. It is also known as negative consequentialism.
Beneficence, on the contrary, says that you must act in a way that is good, or best. You must act in a way that your action brings good or brings more good than evil. If you can do good, you must.
The difference between Nonmaleficence and Beneficence is that Nonmaleficence tells you what you musnt do. Beneficence tells you what you must do.
For example, donating your kidney would likely save someone's life. By the law of Nonmaleficence, you dont have to donate your kidney. You are allowed to do it, but you dont have to. Nonmaleficence commands you not to do evil, but it does not command you to do good or to do whats best. You can do no good, but also do no evil. Or you can do no evil, and only do a little bit of good.
However, by Beneficence, you would have to donate a kidney, since Beneficence commands you to do the greatest good.
Of course, this depends on how good and evil are defined, but you get the idea about the difference between Nonmaleficence and Beneficence.
Now, there is also a third idea, which is Rules.
Unlike Nonmaleficence and Beneficence, which are consequentialist theories that weight good and evil, Rules are moral absolutes that say how you must follow them even if following them produces evil.
For example, if you say "I will always tell the truth", then by that rule you would have to tell the truth even when doing so causes nothing but harm.
Rules, also known as deontology, do not weight good and evil. Rather, they maintain that certain actions are wrong in all circumstances. For example, "it is wrong to lie" means that you shouldnt lie even if it can save your life.
There are advantages to Rules over consequentialism, since Rules are in most cases easier to understand and follow.
For example, if I accept that lying is wrong, then I wont lie in any situation, even when lying would result in something good.
If I accept that violence is wrong, then I will not commit violence in any situation.
Rules dont require us to weight good and evil. Weighting good and evil may seem good and desirable, however the truth is that most people arent capable of weighting good and evil in complicated situations. Therefore, Rules are more suitable for most people.
Rule consequentialism is a form of combination of Rules and consequentialism, where we dont do things that are famous for having bad outcomes, and do things that are famous for having good outcomes. Therefore, Rules would be made according to that which is good in great majority of cases.
For example, "Do not kill", "Do not lie" or "Do not steal" are rules that are good in great majority of cases, therefore should always be followed.
Kant's Categorical Imperative is a way of making rules by simply asking what would happen if everyone did certain action.
For example:
1. Would society be horrible if everyone was stealing things from others?
2. Would society be good if no one was stealing things from others?
Since 1 and 2 are answered with yes, it follows that stealing is wrong. Therefore, the rule "Do not steal" is a universal law that we should always follow, according to Categorical Imperative. Even if stealing would result in good, for example stealing to donate to charity, it is still wrong.
I think this pretty much covers these three moral theories.