Never give a sucker an even break!

There are many meanings for the term 'sucker' not all complimentary. In horticulture a sucker is usually growth coming from a rootstock at the base of a grafted plant. Rootstocks are often used to impart vigour, disease tolerance or sometimes the ability to cope with unfavourable growing conditions. A strong root system will promote vigorous growth. 

Rose varieties  are usually grafted onto a vigorous rose rootstock to impart vigour, fruit trees are often grafted onto a dwarfing rootstock to achieve the opposite effect. Most garden trees are grafted as they will not easily root from cuttings and garden varieties of tree will not come true from seed.
Whatever the reason for grafting a plant, it is important to ensure that the rootstock is not allowed to grow shoots. This is achieved by timely removal of any sucker growth as soon as it appears. 

If possible, I use a pruning knife to remove sucker growth coming from the stem, leaving a 'stub' will encourage regrowth. For suckers arising directly from the roots, it may be possible to pull out the sucker growth directly from the root system. If this is not possible, I detach it using a sharp hoe or spade. 

In some cases the difference between the growth of the desired plant and unwanted sucker growth is clearly evident. Cotoneasters may be grafted onto hawthorns and pears onto quinces. In these instances, suckers will have totally different foliage to the desired plant. In other plants you may have to look carefully at the base of the plant to see the 'graft union' - the join between the rootstock and the scion (desired plant) . Growth coming below the graft union will be undesirable sucker growth. 

Sucker growth coming from the rootstock of a grafted Viburnum. This will need prompt removal.

Suckers of Rhus typhina. Unless removed promptly, the shrub will rapidly become a thicket