Sometimes trees must be felled, always leaving a stump behind. Some people call in a 'stump grinder' who uses power equipment to tear the stump into coarse chips and shreds, which are usually blended into the soil. However, without the right kind of nutrients and growth of bacteria and fungi, it can take a long time for the wood to break down.
You can take a different approach using a drill and high nitrogen fertilizer. Use a drill with a spade bit (3/4 to 1-inch size) to drill holes vertically into the stump, roughly a finger's length apart and a few inches deep. Fill the holes with high nitrogen fertilizer (a balanced fertilizer can work, but it is the N that helps here). The fertilizer greatly increases the nitrogen that microbes need to break down the wood. Depending on the type of wood, decomposition should allow you to more easily chop out the stump with a maul or a pick in several months to a year or so.
Chemically, 'stump remover' chemicals do the same thing, they just cost a lot more.
A variation on the process is to bury the stump with a nitrogen-rich material, such as manure or compost. You can even construct a raised bed around the stump and bury it. The stump will decompose, leaving organic matter behind.