I think the best thing to do, if you havent already, is to prove each distribution function from their assumptions and basic principles in probability. That is what I did.
The Binomial asks for $k$ discrete successes out of $n$ discrete chances.
The Poisson is essentially the Binomial. Except that the Poisson is taken over infinitely many infinitesimally small $n$ chances. Basically, we are asking for $k$ discrete successes over a continuous variable, such as time or distance.
They are easy to distinguish by asking what kind of variable are our successes (discrete or continuous) and what kind of variable is the domain it occurs in (discrete or continuous).
The hypergeometric is, oddly enough, also similar to the Binomial. Except instead of two discrete possibilities (success or fail), there are more than two discrete possibilities (such as sets A,B,C). When one of these sets contains no elements at all, you are left with the binomial.
The negative binomials is... based on the binomial. Coincidence? The only difference is that we are not asking for the probability of $k$ events in $n$ chances. Rather, we are asking for the probability that it will take $n$ chances to achieve $k$ successes.
I really dont know how to answer your question except to compel you to learn the basis for each. How each was proved from simpler probability concepts. All the ones I explained are fundamentally based on the binomial. But the equations are simplified from certain assumptions made about the problems. If you understand these assumptions you can run through the proofs yourself, and you will surely understand how to use each distribution.