Duotone images are simply those with only two tones. Tritones have three. Instead of lots of greys or colors, you can boil down an image to simple expressions of two or three forms of expression. Sometimes the effect gives a better image.
The images above use only three tones – black is not one of them. Essentially I’ve selected three versions of grey and Photoshop has replaced all those RGB or B&W pixels with some form of these three. If color gets in the way of an image then why can’t the almost infinite range of tones from black to white? How about just eliminating all that and use just the bare minimum needed to display luminance, contrast and shape?
This is not to be confused with toning an image. There are selenium tones, sepia tones, cyano tones, colored tones, etc. Doing that essentially drapes that tone over the whole image and the colors (or greys) showing through are influenced by it. A duotone or tritone doesn’t drape the tones over the image – they become the image.
I’m sure there are lots of sites giving technical background on this and I could be totally wrong about it. The idea goes back to graphic arts and halftone printing; it’s been upgraded to the digital world. I’m sure experts in the art can turn a drab image into a spectacular one – I’m just playing around.