Deriving Taylor and Maclaurin Polynomials

I am just reading about Taylor and Maclaurin Polynomials. I was wondering how they were derived. Could someone please show me how in both a rigourous mathematical way and an intuitive way. I hate to just use formulas that I don't have at least some grasp of the theory behind them. As a toddler my parents gave me a set of real tools (yes they were insane) and I comenced dismantling everything. I took apart tables, telephones and finally the TV, which put an end to my tools.

Anyway I appreciate the insight from whoever, the more the merrier.

ManyArrows