ASCII was (and still is) just a simple set of conversion rules to go from numbers to characters.
Unicode was the solution to an increasingly important problem in the dawn of computing and the internet: How does my computer communicate with another computer on the other side of the world if that computer 'speaks a different language'? One of the most popular 'languages' in the early 1980s (especially in the USA) was ASCII - the American Standard Code for Information Interchange. It's the organisation that handles the international standards for converting numbers into textual characters.
Okay, now on to the long explanation: The long explanation starts with an international organisation called 'Unicode'.
