Uncorrelated and zero mean image noise
Let us consider two random variables $X$ and $Y$. You are aware of the following terminologies with respect to (wrt) the random variables:

(Zero Mean) : $\mathbb{E}(X) = 0$ and $\mathbb{E}(Y) = 0$,

(Uncorrelated) : $\mathbb{E}(X Y) = \mathbb{E}(X) \mathbb{E}(Y)$. It is interesting to figure out how it is different from independence.

Now, an image specified, say $g(x,y) = f(x,y) + n(x,y)$ is a discrete random random process. The notion of zero mean and uncorrelated are not very different here, as for any two points $(x,y) = (x_1,y_1)$ and $(x,y) = (x_2,y_2)$, the image intensity at these pixel locations are random variables.