Ombre refers to the gradual blending of one color to another or a gradual change of light color to dark.
This term is used in textile printing to refer to dyeing the yarn or fabric to this color effect or printing on the fabric surface to get this effect. Fabric can be woven with dyed yarns in the ombre shading. You can also do embroidery with the same effect.
The English word is taken from the french word ombré which means shade; à l’ombre de means ‘in the shade of’.
(Ref : Dictionary)
The colors graduating from a light shade to a darker shade have a very soothing and aesthetic effect. You can use monochromatic colors of a single color to get this shaded design. The dark color fading into a lighter shade has a subtly dramatic element to it. When the colors fade from one area to the other, it adds a depth which is not there if it is a plain color.
Different colors which look good when placed alongside each other can also be shaded in this way to get the ombre shading. You can check out this post on color combinations to see which colors appeal to you.
In makeup, the term is usually referred to talk about the fashion trend for hair dyed in this way.