In the poisson case you set the sample mean equal to the population mean getting...
The PMF of the Poisson distribution is given. E(X) = Lamda and Var(X) = lamda.
Find the method of moments estimator for: lamda with a "^" on top = lamda.
Sorry but I don't know how to add mathematical operators in the post so it would be easier for you to read!
In general, with the method of moments, find the relationship between the moment ( ) and the parameter .
Let's say you have
Then define
You'll have and that's all...
Sometimes, you may have a multivariable function :
In which case, it's exactly the same reasoning.