(1) A language is considered natural if it evolved along with a culture of human native speakers who use the language for general-purpose communication. Languages like English and Japanese are natural languages, while Esperanto is an artificial language, having been deliberately constructed.
(Artificial languages can still have native speakers, if children learn it at a young age from parents who have learnt the language. Thus it is possible for there to be native speakers of Esperanto, for instance, though the number of native Esperanto speakers in the world is probably quite small if there are any at all.)
(2) Sometimes any language used by a human being (be it naturally evolved like English, or artificially constructed like Esperanto) is considered a natural language, while programming languages are classed as artificial. This usage can be seen in the term natural language processing.