Holistic is a term traditionally used to describe any approach that deals with the 'whole'. In holistic medicine, for example, the idea is that all of the aspects of health including psychological, physical and social are so intimately connected that investigating and treating any individual condition, system or organ is only possible by reference to the whole.
By its very definition, a dog food cannot be holistic since it is only one of a number of factors that contribute to a dog's health. Nevertheless, over the last couple of decades, more and more pet foods have been labeled as 'holistic'. Originally, the term was most often used to describe foods that aim for general health throughout the whole animal as opposed to foods that target specific issues like the coat, teeth, joints etc. This is a commendable approach but by this definition 90% of foods would be holistic.
Nowadays, the term has lost virtually all of its meaning and is used arbitrarily throughout the pet food industry. Often it is used as a byword for natural or hypoallergenic but since there are absolutely no criteria for describing 'holistic' foods, any pet food manufacturer can legally describe their products as holistic no matter what their ingredients or their properties.