Trianglated Category and Derived Categories

Trianglated categories and derived categories are two thing I heard but do not learn for sake of short in motivations.

In this semester, I take a course called homological algebra and I was asked to talk about trianglated categories and derived categories, so I just want to know what is it by this chance. What's more, I draw as much comm. diagram as I can to illustrate the main point, I think you can read it even if you cannot distinguish left-right or row-column (like me). Here is the notes on them---forgive my poor english gramma and the numerous typos.