What is meant by thresholding and image segmentation? How can a histogram representation of an original microarray image be used to perform both these procedures with the goal of ‘automatically’ identifying microarray spots in the original image?

Thresholding is an old, simple, and popular technique for image segmentation.
Image segmentation is to divide the image into disjoint homogenous regions or classes, where all the pixels in the same class must have some common characteristics:

GB (x,y) =
{1, if G (x,y) >T
{0, if G (x,y) ≤T

Where:

G(x,y): The input gray image that we want to segment.

GB: segmentation result. Forms a binary image, in which each value of GB(x,y) gives the category that the corresponding pixel belongs to.

T: Threshold value. It is an integer within the range [0...K]

G: Image.

The pixels of a microarray spot can be regarded as either just showing background noise or real signal. In a histogram of intensities (often log-transformed values are being used here) this is reflected in a bi-modal structure (camel-like curve). Statistical models can be used to define a threshold between the two modes and thus classify pixels as "background" or "signal".