What is Tolerance Stack up analysis? Please Explain

Tolerance stack up analysis is a mathematical exercise where the interfacing features between mating components of a design are adjusted to their worst case size and location tolerances to determine if they will still be able to assemble in that condition.

It is one thing to design a widget.

It is entirely another thing to design a widget that can be cost effectively mass produced to a six sigma level, and tolerance stack up analysis (among many other things) is very necessary in this case.