Uncorrelated noise - a noise without any correlation between pixels.
Correlated noise - a noise that has a correlation between different pixels or time correlation in the same pixel.
It is possible that a certain image pixel will be deviated by both correlated and uncorrelated noise. The mathematical modeling of this is a summation of the two noise processes.
Noise - In image processing, a deviation of the value of a pixel is called noise. An image is basically a table of values. A grayscale image is a single table with brightness values per pixel while a color (RGB) image has three tables, one for each color. Suppose that a single-pixel (grayscale for simplicity) is supposed to hold the value 119 but for some reason, the value changed into 121. There are many reasons why this could happen but commonly, we are discussing causes that affect many\all pixels. This will distort the image with respect to the severity of the cause.
Correlated noise - A noise that deviates the values in a correlated way. The correlation is pixel-wise. For example, think of the image acquiring process. the light goes through a lens in the process which has a shape designed to concentrate the light. Obviously, manufacturing methods are not perfect and some error occurs. The results would be a small smear of the image. If you will take a picture of a black point on a white background with such a lens it will spread into other pixels on the image plane, that is, instead of all pixels being 255 (white) and a single-pixel being 0 (blank), neighboring pixels will also have small values (10?20?). This is called the point spread function (PSF) of the lens and it is an important parameter we wish to be small. There will be a correlation in the noise, between neighboring pixels. In PSF it is modeled as a 2D gaussian:
The center will be most affected (red) and as we get farther away the effect will fade (blue).
Note - Correlation could be with pixels that are far away from each other (sparse correlated noise) or could refer to time correlation, a correlation of a single-pixel noise over time (in a video).
Uncorrelated noise - A noise that has no correlation between pixels. For example, deviations in the manufacturing process of different picture elements (pixels) can cause each one of them to record the light it is experiencing a little differently. For example, think of tow neighboring pixels that experience the same light intensity (we assume they supposed to hold the same value). One will record a certain value and the other will record a slightly different value. Current pixels are electrical components of a CCD which may record the light slightly different from one another due to differences between them. The way in which they differ could be an intuitive example of a random process that holds no correlation between two pixels. Such noise is uncorrelated.
Note - it is not a good example because the manufacturing process of CCD (photolithography) will probably have a correlation between 'messing up' neighboring pixels due to its nature. But still, it is a good example to understand uncorrelated noise. Another example could be an image wich was corrupted while sent over a communication channel (satellite?). Let us assume that the communication corrupted each pixel in an uncorrelated manner, hence, uncorrelated noise.
The different processes which cause the deviations of the values are acting together in a superposition manner. Is simpler words, you can calculate the effect of each noise separately and then sum up all the effects. This means that a pixel could have both correlated and uncorrelated noise.