I have not done much plant grafting. During my undergraduate studies I grafted pink geranium shoots onto a white geranium, resulting in a rather ugly specimen. Most of my experience with grafted plants has come in the role of caring for plants that have been grafted in nursery and landscape settings. When plants are grafted there are two parts: a rootstock and a scion. The rootstock, which may also be called an understory, that is the part of the plant that has the original root system. Many times the rootstock is selected for disease resistance or strong growth. The scion is the shoot that is grafted onto the rootstock; it is usually selected for ornamental value. Since grafting is labor intensive and requires special skills it is only done in the commercial market when it makes economic sense. There are many reasons grafting is conducted, but the dominant reason is to asexually propagate a plant with desirable characteristics. This is particularly true in the case of hybrids and genetic oddities, when sexual reproduction is not possible or would result in loss of desired characteristics.
After a graft has healed and the vascular tissue has been joined, the primary task in plant management is to keep an eye on the rootstock. Many exceptional ornamental plants are produced by grafting a scion from a hybrid onto a rootstock from one of the parent species. A rootstock might produce shoots that compete with the scion, and if left unsupervised, might overtake the scion. Here is an example from a looming problem on a hybrid witch hazel I saw today:
The little shoot rising from the rootstock must be terminated. It will have inferior ornamental attributes compared to the scion and will likely be more vigorous. Right now the uprising can be quelled in a few seconds, one quick cut with a pruners. I posted a picture of a leaf from this particular hybrid witch hazel earlier this fall. The scion has dramatic colors, while the rootstock shoot will be a weak yellow.
As a horticulturist I notice grafted plants in landscapes that have reverted to, or are in the process of reverting to, the rootstock. What appears to the untrained eye as a peaceful looking plant might in fact be two strong-willed, competitive individuals functioning as one organism.