In general, gray is between black and white. Let's say we digitize them 0, 1 and 0.5. In addition, it would be meaningful if we specify the size by using the general formula of 2^k (2 power the scale you need).

Black and White image contains only two levels.

Gray image represent by black and white shades or combination of levels for e.g.

8 bit gray image means total 2^8 levels form black to white 0 = black and 255 is White.

Grayscale is a range of shades of gray without apparent color. The darkest possible shade is black, which is the total absence of transmitted or reflected light. The lightest possible shade is white, the total transmission or reflection of light at all visible wavelength s.

A black and white image is one in which at each pixel position there are just two intensities, full intensity i.e. 1 and no intensity i.e.0. If we allow pixels to have intensities in range like 1, 0.9, 0.8,...,0.1,0 , such an image will be called a gray scale image. However, these total number of steps in this range are in power of 2 (8,16,256 etc.) to make them suitable for digital systems.

grayscale image have a continues range of gray values while a binary image is a digital image that has only two possible values for each pixel.

Grayscales are represented as integers within the computer. In order to represent a gray scale by a number, a computer must first decide how many bits (binary digits) to use to represent the grayscale.

Luminance is coded on one bit for B&W ... or more for Gray (8 in general). In colored frames a pixel is 24 bits : 8 for Y (Luminance), 2x8 for color differences (U and V)

Usually, we call B/W image to the panchromatic images. These images have 0-255 DN (8 bits images) that usually are represented using 255 gray values (gray scale images in technical words but B&W images in TV denomination.

http://presentationsoft.about.com/od/g/ss/120307-grayscale-picture-definition.htm

http://answers.yahoo.com/question/index?qid=20080703035011AAhLziR

Gray scale image have 256 level of light in image (0 to 255), if you use only 0 and 255 gray level, you have a black and white picture

grayscale image has the value from 0-255, whereas black and white has the value of 0/1 only. 

Grayscale image has gray values ranging from 0-255 where 0 =black, 255= white while the black and white image has only 0 and 1 values where 0 =black, 1=white. 

Black and White image is a special case of Grayscale image where the gray value/level is restricted to be pure white or pure black only in contrast to varying gray level of white or Black in case of gray scale image.

Binary images are images whose pixels have only two possible intensity values. They are normally displayed as black and white. Numerically, the two values are often 0 for black, and either 1 or 255 for white.

Binary images are often produced by thresholding a grayscale or color image, in order to separate an object in the image from the background. The color of the object (usually white) is referred to as the foreground color. The rest (usually black) is referred to as the background color. However, depending on the image which is to be thresholded, this 📷polarity might be inverted, in which case the object is displayed with 0 and the background is with a non-zero value.

Some morphological operators assume a certain polarity of the binary input image so that if we process an image with inverse polarity the operator will have the opposite effect. For example, if we apply a closing operator to a black text on white background, the text will be opened.

Reference:

https://homepages.inf.ed.ac.uk/rbf/HIPR2/binimage.htm

https://homepages.inf.ed.ac.uk/rbf/HIPR2/binimage.htm