Recommended Reading with Nice Images:
Radiographic Techniques, Contrast, and Noise in X-Ray Imaging
Contrast can be thought of as the percent difference in values between two things in an image. The bigger the difference in value between two objects, the greater the contrast. For example consider a white square (thing 1) on a gray background (thing 2). If the brightness value of the background is 10 and the brightness value of a square is 25, then the contrast is (25-10)/10 = 1.5 or 150%. If contrast is 0, you cannot detect a difference between the two objects.
Noise is the amount of variation present when the signal should be exactly the same. Noise is random and a simplistic way of measuring it is to take the standard deviation of the brightness values. As noise increases, things become harder to detect.
Contrast-to-Noise Ratio (CNR):
CNR attempts to give a single number combining the effects of contrast and noise. CNR is typically a relative thing, thus we say if CNR increases it means that it will be easier to detect the difference between two things and if CNR decreases it will be harder. If the contrast between things is large, then you will still be able to see a difference if there is a lot of noise. If contrast is small, then noise must also be small in order to detect the difference between things.