Depending on the size and age of your maple tree, there is a good probability that the tree will resprout from the stump. Applying herbicide to a freshly cut stump is the best method to ensure the stump won't resprout on you in the future. During the growing season (leaf-on), cut the tree flush at the ground level and apply glyphosate (e.g., Roundup) to the fresh stump [read and follow the directions on the herbicide label].
It is important that you use a glyphosate product that has ~ 41% active ingredient. The $8 spray bottle of Roundup you buy at the local garden center likely won't provide the results you are looking for because the % active ingredient is too low; therefore shop around to the find the appropriate glyphosate product.