Hi
You really need to decide what tree it is. Most broadleaves will sprout from the stump, most conifers will not. Conifers that will are Redwoods, yew, and some others that do not come to mind at the moment.
Roundup at 20% in water brushed on all exposed bark will kill it.
Your local hire centre should have a stump grinder that will get in a garden and do the job, at a cost. The cheapest is to persistently cut off new growth and it will die, but you have to do it when its leaves not twigs turning into branches.
You did not say what diameter the stump is? A mini digger would also do it.