Digital image moments in plain English
An image moment is simply a number that characterizes the image, construed as the realization of a spatial random variable. If you've taken any probability class, you should remember the concepts of mean and variance, which are derived from the first and second moments of the random variable (the n'th moment of an r.v. is the expectation of its n'th power). Furthermore, the moments of a random variable collectively prescribe its distribution. In other words, you can reduce a probability distribution to a sequence of numbers, and this is useful when you want to compare distributions numerically.

In image processing, if you want to compare images, you might not want the comparison to be sensitive to minor things like rotation, translation, and scale (since the image remains fundamentally the same). Thus the motivation behind invariant moments you see in the Wikipedia article you cited.