Answer: Suckers can be different things on different kinds of plants. Without knowing which plant you are concerned about it is difficult to be specific. But, in general, they are small shoots or new growth that are forming where you would not want them.
On a tree, a sucker can grow straight up from the roots and compete with the main trunk. These are removed as soon as they appear. Sometimes they can be pulled/twisted off, sometimes they have to be cut.
Some shrubs will naturally sucker from the roots or stolons, sending up new shoots or branches from the ground all around the perimeter of the plant. If allowed to grow, the suckers will develop into new shrubs and spread to create a colony or thicket. Sometimes this is desirable, such as in an informal setting, while other times it is not.
On a tomato plant, a sucker is a small branch that begins to grow from the main stem in between two older, larger branches. This are often removed by pinching them off with your fingers or trimming them off with a hand pruner. The purpose is to thin the plant and allow air and sun to reach the lower growing fruit as well as to prevent foliage diseases. It also reduces the weight of the plant being supported by the trellis.
I hope this helps.
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