These are two horticultural terms that mean almost, but not quite, the same thing.
A sucker is a root sprout: it grows from a bud on the root, often quite a distance from the mother plant.
An offset is a basal shoot: it grows from a bud at the base of a plant.
Both are genetically identical to the mother plant.
Both Good and Bad
Both can be seen as something positive or negative. You might appreciate an offset or sucker or two from a favorite plant that you could then dig up and move elsewhere. And where would gardeners be if short-lived plants like raspberries (each plant only produces one stem and one crop, then dies) if they didn’t produce suckers and thus keep the raspberry patch going from year to year?
However, suckers especially can be annoyingly invasive. Raspberries also go too far when their numerous suckers invade neighboring beds and even within the raspberry patch, need to be thinned out. False spirea (Sorbaria sorbifolia), staghorn sumac (Rhus typhina) and common lilac (Syringa vulgaris) are other plants that annoy gardeners by suckering too widely.
Another problem appears when the plant you’re growing is grafted, that is when a desirable plant (more floriferous, more productive, with a more attractive habit, etc.) grows on a rootstock of a less desirable plant. But if the rootstock produces offsets or suckers, the resulting plants will resemble the “ordinary” rootstock rather than the desirable plant. These unwanted growths can come to dominate and even eliminate the desired plant, so don’t wait, cut back any growths from a rootstock as soon as you recognize it. This problem of offsets and suckers arising from rootstocks is common in fruit trees, standard (top-grafted) trees and bush roses, all of which are produced by grafting.
Some plants routinely produce offsets or suckers, but others only do so if the health of the plant is in some way threatened, for example by disease, physical damage or pruning. You can often promote suckering by cutting a plant back severely, nearly to the ground: this is called stooling.
Other plants only produce suckers from their roots when the roots are damaged in some way. Chopping into the ground around the plant with a shovel or just deep hoeing may stimulate suckering.
To Remove a Sucker
If you decide that a sucker doesn’t interest you and you want to eliminate it, tearing it off is often more effective than cutting it back, as pulling is more likely to damage the bud from which it sprouted. If you do decided cut to cut it back (and some suckers simply can’t be yanked off), remove some of the soil and cut it back right to where it joins the root or even cut off that section of root. This can help keep the root bud from resprouting.
You can also use herbicides to control suckers.
If a plant has a reputation of sucking a lot, you can surround it with an in-ground barrier such as rhizome barrier (bamboo barrier) or a pail with the bottom removed to keep it in check.
What About “Stem Suckers”?
Some plants – apples notably – produce unwanted growths from latent buds on their stems, branches or trunks that gardeners sometime refer to as suckers. In fact, though, they aren’t suckers because they don’t grow from roots. They’re are correctly referred to as water sprouts, not suckers. Water sprouts tend to have a vertical, upright growth pattern and rarely produce flowers or fruit. They can simply be pruned off.
As for the so-called “suckers” on tomatoes, they don’t grow from the root system and therefore are not suckers either. They are, in fact, just normal branches, secondary stems that grow from the leaf axils. Whether or not you let tomato branches grow and produce fruit or remove them so you can better attach the plant to its stake is entirely up to you.