A sucker is an unwanted shoot from the base, as opposed to a good basal shoot. There are two kinds,

1. On grafted roses, a shoot from below the graft coming from the rootstock, which is a different rose variety.

2. On ungrafted roses, a sprout from the roots some distance from the crown of the plant as the rose tries to turn itself into a thicket. Most modern roses don't do this, but old Gallicas and some others commonly do.

Rugosas are notorious for suckering.

I plant my rugosas and gallicas where they can sucker to their hearts content and form lovely thickets the deer won't eat. When they do get out of line, I pull them out and pot them up and give them to friends who want lovely thickets...