Andrew, it's the process by which a rose was spliced onto rootstock from a more vigorous rose. Sometimes the whole rose (known as the scion) is grafted onto the rootstock, and sometimes a bud eye or several bud eyes from the scion are grafted onto a rootstock. The two different methods were once differentiated as "grafting" and "budding," but today the terms are virtually interchangeable.
This was once done to all roses because it was thought that the addition of a vigorous rootstock was necessary for all roses. The high cost of grafting services, the shortage of experienced grafters, and other economic considerations gradually led to the sale of own-root roses, which are rooted cuttings from a rose. Grafted roses typically were grown in the fields for two years before sale. Own-root roses, by contrast, usually are sold as soon as 6 months after the cutting has rooted.
Now grafting is done mainly on large-flowered climbers, grandifloras, and hybrid teas, which tend to grow poorly or not at all on their own roots. Most of the old garden roses, floribundas, and shrub roses will grow well on their own roots and do not require grafting.