Suckers are branches that grow from the base or root of the rose bush. Many varieties of roses are grafted onto resistant root stock. The root stock will send out branches that are of the original variety, not the variety of the graft. These branches, called suckers, will eventually take over the plant and crowd out the original bush if left on the plant. Remove any branch growing from below the graft.
Watch for new branches sprouting from below the graft. Remove the branch with sharp cutting shears, cutting at the trunk.
Look for new plants or branches growing up from the soil around the base of the plant. Do not cut it at ground level, it will come back even more vigorous than before.
Remove the soil around the sucker, following it back to where it attaches to the main plant. Break the sucker off at the point of its attachment to the plant. Remove it and its roots.
Plant the sucker, if desired. It will be genetically identical to the rootstock. Transplant it to a new location with plenty of room to grow. For roses grown on their own rootstock this is a good way to propagate the plant.