While there is no permanent way to remove algae, there is one product that appears to out-perform others. It's called Fountec.
I would recommend a complete cleaning of the fountain before the initial application of this or any other algaecide. You don't mention what the composition of the fountain is....concrete, cast iron, fiberglass, polymer, etc.. If you still have the owner's manual for the fountain, check the recommended cleaning do's and don't's. You may nee to start with a chlorine solution. If any problem areas remain, these can be soaked with full strength vinegar for about 10 minutes (Use a saturated rag or paper towels placed over area to be treated). It is important to ascertain that the vinegar will not react with the fountains surface. Rinse and allow the fountain to thoroughly dry before refilling and adding the algaecide.
Unfortunately, your fountain may require weekly dosing of algaecide in the hotter months and less often in the winter. Alage is usually with us all year here in Florida, but explodes in the longer hot days of summer.