Here's much easier way to understand it:
You can look at Binomial distribution as the "mother" of most distributions. The normal distribution is just an approximation of Binomial distribution when n becomes large enough. In fact, Abraham de Moivre essentially discovered normal distribution while trying to approximate Binomial distribution because it quickly goes out of hand to compute Binomial distribution as n grows especially when you don't have computers (reference).
Poisson distribution is also just another approximation of Binomial distribution but it holds much better than normal distribution when n is large and p is small, or more precisely when average is approximately same as variance (remember that for Binomial distribution, average = np and var = np(1-p)) (reference). Why is this particular situation so important? Apparently it surfaces a lot in real world and that's why we have this "special" approximation. Below example illustrates scenarios where Poisson approximation works really great.
We have a datacenter of 100,000 computers. Probability of any given computer failing today is 0.001. So on average np=100 computers fail in data center. What is the probability that only 50 computers will fail today?
In fact, the approximation quality for normal distribution goes down the drain as we go in the tail of the distribution but Poisson continues to holds very nicely. In above example, let's consider what is the probability that only 5 computers will fail today?
Hopefully, this gives you better intuitive understanding of these 3 distributions.