Most garden roses have been budded on to a vigorous rootstock system, which to all intents and purposes is not seen and is not generally visible as part of the plant.
The exception is if the rose was planted a little high in the soil, or if there has been some erosion of soil around the root system. In this case, a brown 'stub' can be seen at the bottom of the bush, protruding from the soil. (The rootstock is from one of the vigorous Briar type wild roses, and has no other purpose than to support the choice rose which is budded onto it.)
Increasingly, roses are being grow by other methods - which do away with the rootstock - and also the suckers!
The new basal growth of rose stems also emerge from the rootstock, but from the top.
It is from this root stock that rose suckers can emerge. The suckers are simply stems forming from the vigorous rootstock. These suckers are much more vigorous than the cultivated rose, and will soon rob the choice rose of its food supply unless dealt with.
With bus roses, the suckers emerge from ground level or below. With standard roses, suckers are sometimes seen emerging from the 'standard' stem.
The dark stem at ground level is the rootstock. It is from this rootstock, and the rest of the root system that suckers emerge.
Suckers normally emerge un-noticed, then suddenly seem to shoot above the rest of the rose bush in a matter of weeks. They are quite easily distinguished from the real rose, in that they have 7 leaflets making up the leaf, are very thorny, and also generally have light green and slightly ribbed foliage, as distinct from more bush roses foliage which is glossy and a darker green with 5 leaflets making up the leaf.
The picture clearly shows the sucker growths - seven leaflets, slightly ribbed leaves and very thorny stems.
They are best dealt with by pulling and twisting them off - rather than cutting them with secateurs. You may have to remove some of the soil around the root on bush roses to get at the source of the sucker. The sucker can emerge several feet away from the main bush - sometimes in lawn areas! Make sure that you wear thick leather gloves for this task, for the fine thorns can prove difficult to remove from your skin - flesh!
Suckers are generally caused by damage to the root stock - either by forking or other cultivation damage, being scuffed if the root system is near to the surface, or as is the case in the lawn, where the rose has been planted near the edge and the root system gets 'scalped' with the mower.
Miniature roses and many modern roses do not suffer from suckers, as they have been produced from cuttings rather than being budded onto the wild root stock.