Basic question about the normal distribution

So it seems like different distributions have different kinds of "experiments" associated with them, like how you use the binomial distribution when looking for the probability of k wins out of n trials with probability p. What I'm wondering is what kind of experiment is associated with the normal distribution, and what values you need given to you before you can use it (like, I know you use the Poisson distribution when you have a binomial-type distribution with an infinite number of trials, but you already have the expected value).

I'm also interested in finding out how the p.d.f. of the normal distribution can be derived from the p.m.f. of the binomial distribution, if anyone has a link to a proof or something.
I don't understand how the binomial distribution "approaches" the normal distribution as the number of trials goes to infinity. Isn't the "hump" in a bar graph of the binomial distribution at least in part a result of the finite number of trials?