Part-of-Speech tagging is the process of reading natural language text and assigning parts of speech to each token.
One could imagine taking in a sentence like:
The dog ran away.
and creating a data structure that had the following annotations:
The[article] dog[noun] ran[verb] away[adjective].
Words can have different parts-of-speech depending on their context. For example, the word away can be either an adverb or an adjective, or part of a larger phrase.