This is named Pascal's triangle in the western world after Blaise Pascal although it has been found earlier than him. (China used it hundreds of years before him)
Here are some quick pics:
Forming the triangle:
Patterns – the triangle numbers in one pair of the diagonals:
Patterns – the numbers in each row add to a corresponding power of 2:
(The sum of each row is double that of the last – this is effectively because each number in every row 'contributes' to two numbers in the row below it)
There are more patterns too
Video – thanks to @MR.GAM3R!