Bernoulli number : why are they defined as the coefficient of the Taylor series of $\frac{y}{e^y-1}$? Why Bernoulli number are defined as the coefficient of the Taylor series of $\frac{y}{e^y-1}$ ? In other word, why the coefficients of the series of $\frac{y}{e^y-1}$ are so important ?
 A: Bernoulli numbers can be defined in a number of ways in accordance with the myriad of places they show up in math. See: https://en.wikipedia.org/wiki/Bernoulli_number . 
That being said, I think that question "Why are the coefficients of the series of $\frac{y}{e^y - 1}$ so important?" has the simple answer: "Because they are the Bernoulli numbers". A more interesting question I guess is what is so special about this function that its coefficients are given by the Bernoulli numbers. Why would the Bernoulli numbers put into a series come out to something like this?
A: Your question should simply be: "Why are the Bernoulli numbers, which are merely the coefficients of the Taylor series of $\frac y {e^y - 1}$, so important ?". Their ubiquitous apparition in "a myriad of math places", in @user357980 's words, makes a slick answer impossible. However, if we restrict to number theory, I think that the most striking - and utterly deep - application concerns the special values of the zeta function; see e.g. the second part of https://math.stackexchange.com/a/1771242/300700. 
Note the following intriguing calculation of the value $Z := \zeta(-1)$ in Ramanujan's Notebooks:
 $Z = 1+2+3+4+... , 4Z = 4+8+12+..., -3Z = 1-2+3-4+5-6+...= \frac 1{(1+1)^2} = 1/4$, hence $Z= -1/12$... which is the correct value !
