What is the best intuition behind the unique parameter $\lambda$ in the Poisson distribution?
Poisson RV is commonly used for modelling number of occurrences of an event within a particular time interval. And, since $E[X]=\lambda$, its unique parameter is referred as mean number of event occurrences within our particular time interval.
Since the Poisson distribution is defined as
It is not hard to show that $E[X]=\lambda$ and $D^2[X]=\lambda$. Therefore you can directly interpret $\lambda$ as the expectation or as well as the variance of the poisson distributed variable.