The rootstock competes with the grafted rose by sending up suckers which draw energy from the rose and these can eventually take over entirely. You need to insure that this doesn't happen by rubbing off suckers where they originate on the rootstock. Ideally you want the rose to grow on its' own roots so it doesn't have to always depend on the rootstock and the bud union. If you plant the graft down two inches or so a rose that is vigorous may create a hairball of roots on the stem above the graft which will be its' own roots and these may develop and eventually takeover so that the rose is on its' own roots.

Many roses sold in the US are grafted onto another rose used as rootstock. The most common rootstock used here is a once-blooming climber names Dr. Huey. It has red blooms, and you'll see them all over. Say you buy a grafted Tiffany. If the grafted rose starts to sucker, that means that Dr. Huey has started actively growing. It will send up canes and eventually grow and kill off the grafted variety (your Tiffany). Instead of pink Tiff,you get dark red Huey. Not good. So, you have to dig down and remove the suckers as soon as you see them, otherwise, you'll end up with a nice plant of the Dr.

If your rose is own-root, then suckers are OK and don't have to be removed, because they are new canes of your wanted plant. Some roses are vigorous, and suckers are removed to try to contain the rose at a certain size.

Most roses sold at local garden centers are grafted on Dr. Huey rootstock. Some may be own-root, it is usually noted on the tag. For online vendors, they usually tell you somewhere on the site if they are own-root or grafted. If not, then ask.

The problem of suckers is a problem affecting grafted roses.
Sorry if I do mistakes in the language, it's not easy for me to write using quite specific terms for plants. Anway, the explanation is very simple: a grafted rose is a rose where on a rose used just as a "base", as a support, one grafts the kind of rose you see. The suckers are part of the "base" trying to overwhelm the grafted rose, and they draw energy from the grafted rose. If you leave them, you will soon have a different rose: not the one you bought but the one coming from the "base".

Exceptions:
1-own roots roses do not make suckers as there is no "base" and no grafting. That's a reason why they are becoming increasingly popular here in Italy.
2-rugosas produce suckers that can be left without any problem, as these suckers are rugosas as well: rugosas grow precisely this way, suckering freely and forming large shrubs.

A sucker is an unwanted basal shoot. Generally, basal shoots are highly desirable, but sometimes beginners get confused and remove them out of sucker-phobia.

There are two types of undesirables: (1) shoots from the rootstock, if a grafted plant and (2) shoots distant from the base of the plant which colonize other plants' space. The second type is common with some types of old roses.