The variation token makes it so that if you have a horse that has a variation for a white pattern (like sabino, tobiano, overo, leopard, etc) if you equip it then breed them they have a higher rate of passing the variation gene to their foal. Note that it only works on two horses that you own.

To tell if your horse has a variation you can inspect the image and look at the layer image. Anything that has a number 2 on a up, is a variation. Anything with no number or 1 is the default pattern variation and thus would not be worth using a variation pass token on. 

Hope this helps explain it a bit better!