1. A natural language is one that evolved along with a culture of human native speakers who use the language for general-purpose communication. Languages like English, American Sign Language and Japanese are natural languages, while languages like Esperanto are called constructed languages, having been deliberately created for a specific purpose.
Constructed languages can still have "native" speakers, if children learn it at a young age from parents who have learned the language; there are estimated to be 200-2000 native speakers of Esperanto, for example.
2. Sometimes any language used by human beings (be it naturally evolved like English, or constructed like Esperanto) is considered a "natural" language, while programming languages and data representation languages for computers are classed as artificial. This usage can be seen in the term natural language processing.