The Little Engine that Could is the title of a children's book originally published about 60 years ago, written by Watty Piper
and famously illustrated by George Hauman
The story is loosely as follows: the Little Blue Engine (train) is carrying lots of toys and goodies to good boys and girls, just puttering along, until it comes to a massive mountain, seemingly too big for the little engine to surmount. The other, bigger, prouder engines say "I can not, I can not" when posed with the feat of climbing the mountain.
In a Herculean feat of self-motivation, the Little Engine successfully climbs the mountain, chanting "I think I can, I think I can". Behold, the power of believing in yourself! : )
The story, the name of the book, and the mantra the book made famous are so well known that the name has entered the common bank of knowledge.