Organic Methods
There are many effective organic methods to kill the ants and get rid of their mound without damaging your yard or the environment. Pour club soda into the mound; the carbon dioxide in the soda displaces the oxygen, causing the ants to suffocate. Boiling water poured directly in the mound will quickly kill the ants in the nest, though it often takes multiple attempts. Equal parts sugar and baking powder poured liberally into and around the mound works by expanding in the ants' stomachs after they consume the mixture.
Commercial Killers
Purchase a commercial ant killer and follow the directions on the package. These killers work well but might have negative effects on the environment.
Plan of Attack
A single nest may have multiple mounds. To have the best chance of killing the queen and the nest start with the largest mounds and work down to the smallest mounds. The larger the mound the older the hill, meaning the queen ants are likelier to be housed there.
Effective Control
Keep watch for any new mounds that pop up. Attack them quickly to effectively thwart the growth of a new nest. Continue to keep an eye out and inspect the yard and garden regularly.