Gamut is the complete range or scope of something - 'the whole gamut of human emotion'.
In photography, we work with the colour gamut - the range of colours that a colour system can display or print. This has never been a problem with my printer - it has wonderful colour rendering.
However, I uploaded some images to Spoonflower. It is a site that prints on fabric, wallpaper or gift wrap. I wanted to make sure my prints would come out the way I put them in. So I got busy in Photoshop and tested the colours with the fabric print profiles that Spoonflower provided. Fabrics include many organic cottons, linens, stretch and sport fabrics and real and poly silk.
The gamut warning looks like the first photo compared to the second one. The out of gamut colours show as gray in the display. In this case, when the lavender was adjusted to be within the gamut range, it washed out to a greyish pink. You can also see the pinks, blues and greens were too bright to print as well. So this image isn't a good match right now - it needs conversion to colours that are within the gamut range rather than just adjusting the saturation and hue in Photoshop.
In the next two photos, the greens become more subdued to account for what can be printed on Spoonflower fabrics so these are a better match for printing.