What are the types of law?

That is a tricky question because it depends by what you mean by “types” of law. If by that you mean areas of law or specialization in law, then the answer is, many.
But if types mean sources and normative categories, then one could say there are four types of law: criminal, civil, common, and statutory. 
Criminal is self-explanatory
civil refers to cases that involve civil action only
common as in customary law
and finally laws that are derived from statutes
 

There are at least 20 types of law, and lawyers often specialize on one of these types of law, like family law attorney, or corporate law, business law, etc.
These distinctions are important because when one needs a lawyer, one must a lawyer practicing in the specific area or type of law. Not all lawyers can practice in any area of law.

That is a tricky question because it depends by what you mean by “types” of law. If by that you mean areas of law or specialization in law, then the answer is, many.
But if types mean sources and normative categories, then one could say there are four types of law: criminal, civil, common, and statutory. 
Criminal is self-explanatory
civil refers to cases that involve civil action only
common as in customary law
and finally laws that are derived from statutes