Parsing is the process of analyzing text made of a sequence of tokens to determine its grammatical structure with respect to a given (more or less) formal grammar.
The parser then builds a data structure based on the tokens. This data structure can then be used by a compiler, interpreter or translator to create an executable program or library.
(source: wikimedia.org)
If I gave you an english sentence, and asked you to break down the sentence into its parts of speech (nouns, verbs, etc.), you would be parsing the sentence.
That's the simplest explanation of parsing I can think of.
That said, parsing is a non-trivial computational problem. You have to start with simple examples, and work your way up to the more complex.
