Here is a question without a short answer, but I will do the best I can.
TPRS is Teaching Proficiency through Reading and Storytelling.
It has also been called TPR Storytelling, which stems from TPR [Total Physical Response] a method that involves teaching a language using body movement or gestures to establish meaning.
TPRS was developed by Blaine Ray who was a Spanish teacher in California and has evolved over the years as more people refine this wonderful method.
So what is the method? I refer you to http://www.blaineraytprs.com/pages.php?page=explanationpage
Basically, there are 3 steps.
1. Establish meaning [usually there are three phrases that are your teaching objective for the day]
2. Ask a Story [the teacher, with the help of the students, creates a story around the target structures]
3. Extended Reading [this is reading that helps the students see the structures that they have learned and reinforces what they learned in the story the day before]
What else is there?
Circling [A questioning technique that gives the student repetition on the target structure]
Personalization [Relates the class and the story to the students]
Comprehensible Input [From Stephen Krashen, an idea that students understand everything that the teacher says. When they receive a new word it is immediately established in a meaningful way, most of the time translation on the board]
Whew! That was a very short version. Check it out on the web if you ever get a chance!