Suckers are adventitious shoots that develop from the roots of a tree, like a genetic clone. They are unsightly and divert important energy from the growth of your top stock.
Some nursery trees are seed grown, but most are grafted with a top stock and a root stock. In other words, a sucker coming from a root stock could be a completely different species than what the top stock is.
If the suckers are small, just pull them out, otherwise, cut with pruners and maintain vigilance for any new growth.