What type of laws were codified when Napoleon was in charge? How did the process work?

The Code civil des Français, informally called the Napoleonic Code or the Code of Napoleon, was established by Emperor Napoleon I in 1804. (I've linked an English translation of the full text of the code, if you like reading law codes and have several hours to spend.)

It is actually one of the first examples of a civil law code, which established simple, consistent rules that the government was to follow based on the written code itself, as opposed to a common law code, in which most decisions are made by courts according to past precedent and custom. You can sort of think of civil law as "law written down" and common law as "law by custom". (Most real-world systems are actually some mix of this; the US is generally considered a "common law" system, but we do have thousands of pages of written statutes.)

Prior to the Napoleonic Code, almost all legal systems around the world were based on common law, laid down by judges. During the Middle Ages, these judges were typically feudal lords such as dukes and princes. (Indeed, the word "court" today refers to both judicial courts and courts of nobles because those were once the same thing.) Good judges could lay down very good laws that would adapt to changing circumstances; but it was also easy for incompetent or corrupt judges to make very harmful laws. One of Napoleon's goals was to create a system of law that was incorruptible, or at least much harder to corrupt, by clearly and explicitly stating what the laws were in a way that would require minimal interpretation by judges. This marked a radical shift in the way laws were written which has been influential to this day, particularly in Europe and in countries colonized by France.

The Napoleonic Code was progressive in some ways, establishing equal rights for all adult White men. But it was also regressive relative to the laws established immediately prior in the French Revolution; Napoleon restored slavery and made family and marriage codes much harsher on women. In practice, even the "equal rights for men" were more honored in the breach than the observance; while formal noble titles were wiped away, France remained in a state of very high wealth and income inequality.