New stems growing directly from the root stock rather than the named variety grafted on the top that produces flowers. Left unchecked this growth has first call on the plants sap so can easily take over.
Rose suckers are thin sappy light green stems coming from the root stock. Usually has different leaf patterning and colour from the named variety.
Treatment and control
General tips: Dig into the soil to trace the shoot back to where it emerges from the root. Pull off the sucker from the root and replace the soil.