What do you mean by noise vairance and mean in an image?
Noise is not just a sudden variation in intensity; that could also be an edge of an object, a texture pattern etc. DC bias variations could also be considered as noise.

Noise is mostly an unrelated (independent of the signal) and random signal, most typically, imposed on the true signal of interest. Note that distortion (which depends on the signal) is something else. 

For practical reasons, noise is mostly assumed to be white, and uncorrelated for simplicty but there are various different noise types as well, such as bandlimited (colored) noise, correlated noise etc. 

For a WSS, zero mean, noise source, the variance gives its power. Stated mathematically: The variance of a WSS random process (noise) $v[n]$ is
$$ \sigma_v^2 = \mathcal{E} \{ (v[n]-\mu_x[n])^2 \} = \mathcal{E} \{ v[n]^2 \} - \mu_x^2 $$ 

When the process is zero mean; i.e. $\mathcal{E} \{ \mu_x[n]\} = 0$,  then 
$$ \sigma_v^2 = \mathcal{E} \{ v[n]^2 \}  $$ 

Which is the definition of power of the WSS random process (noise) $v[n]$

Mean of ergodic noise is, its time (sample) average in the most practical sense. Or it's exptected value as ensemble average, in a more theroretical sense.

It's difficult to separate noise from true signal. A class of techniques to reduce noise, relies on statistical methods to characterize and differentiate noise and true signal, and adjust their filter characteristics according to noise variance (noise power) versus signal variance (signal power).

Optimal Wiener filters exemplify a class of linear (LTI) filters that compute minimum MSE expectations of true signals in noise, for given signall and noise statistical characteristics.