In simple terms, legal positivism is the idea that laws are purely based on what the government officially decides. This basically means that law and morality don't necessarily have to be connected in any way. Legal positivism generally values the clarity of exactness over the nuance of societal belief systems. It is in opposition to the idea of so-called "natural" law, which views law as an extension of moral norms that already exist in the society and, essentially, discredits laws that exist in violation of those norms.
According to legal positivism, if a law is written down by an official authority, it should be followed, even if it isn't necessarily justified or ethical. The obedience to laws regardless of moral opinion is seen as necessary to maintain order in a society. There may be a millions slightly different views of morals and ethics, but legal positivists generally think the law should be free of these nuances in order to avoid chaos.
According to many experts, positivism has both advantages and disadvantages. One of the main potential advantages is the clearness of positivism. When laws are defined in an explicit way, nobody has to use guesswork when trying to stay within legal boundaries. Everybody knows exactly what is expected of them, and many people believe clear boundaries help people avoid confusion, which sometimes makes legal systems more just.
There are also some perceived potential disadvantages of legal positivism. The biggest danger may be the potential for oppression by the majority over the minority. For example, there was a time when slavery was technically legal in the United States and many other parts of the world. Some experts would argue that defiance of those kinds of laws is not only justified, but actually necessary in order to move society forward. Another example would be protesters, who often violate the law during various demonstrations, but might occasionally have strong ethical reasons for their violations.
The idea of natural law is generally in total disagreement with legal positivism because it postulates that moral concerns are more important than what is actually written down. In practice, most democratic governments end up enforcing laws in a way that's not quite in agreement with either extreme. For example, the law may require someone to be arrested for some crime that isn't actually supported by most citizens for ethical reasons. Once the person goes to court, however, a judge and a jury may decide to be very light with their sentencing, or even refuse to convict the person at all. In this way, there is a human element of natural law that sometimes serves as a moderation for the potential harshness of legal positivism.