It's called a Babbage Difference Engine. According to the Wikipedia article..
A Difference Engine is an automatic, mechanical calculator designed to tabulate polynomial functions. Both logarithmic and trigonometric functions can be approximated by polynomials, so a difference engine can compute many useful sets of numbers.
According to the comments on the video..
This is a working hand cranked Babbage Difference Engine made using LEGO parts. It can evaluate any polynomial of the form aX^2 + bX + c for X:0 through N with answers to 3 digits.
For this video the machine is set to calculate the squares of the integers. The video shows the computation from X=2 through 8. It will output 4, 9, 16, 25, 36, 49, and 64.
Don't completely understand exactly how it works, thus my intrique!