Here's a Wiki article on the subject. Sadly it doesn't do a good job of explaining the significance of the function.
Of course it may mean different things to different people (for mathematicians it may be important for reasons entirely different from the reasons it is important to engineers). The following is a hypothetical problem explaining the significance I am looking for.
A laser printer has a targeting system where a laser is supposed to mark a paper at say x=6cm. The standard deviation measured on a large number of attempts to mark the ordinate turns out to be 0.1 cm. If say a billion attempts are made to target x=6cm, what is the maximum expected ordinate that will be marked? Does the error function help me in solving such problems?