I am hoping that pictures will do more than words to explain these terms.
Bud Union = The area of the graft sites. After a few years, the area may look like a 'big, knarled lump' at the bottom of a grafted plant. In the case of a rose tree (as shown here), the 'lump' is at the top of a long cane from the rootstock plant.
Graft Sites = The original locations on the rootstock cane, where another rose variety is grafted. The process of grafting involves cutting into the outer layer of the cane and inserting an piece of another rose. (*See footnote below) Eventually, these graft sites enlarge and merge to form a bud union which resembles 'the lump' above.
Basal Breaks = Any new growth which emerges from the graft site base or the bud union. This is the 'good' stuff; and it should be encouraged and protected, especially while it is tender, new growth. It will become a strong, new stem or cane of the grafted plant.
If new growth emerges from the rootstock or roots, that stem/cane would be called 'a sucker'. This is the 'bad' stuff; and it should be removed immediately.
*Note: Grafting is a form of plant propagation. Using patented roses varieties for the purpose of grafting, without
writen permission of the hybridizer/patent holder, is against the law.