Herbicides
Non-selective herbicides and specially formulated cactus herbicides can be bought in most garden centers. They are most effective when used around the base of the plant in order to kill the roots. Take additional care to prevent harming any nearby plants you may wish to keep. When the cactus is dead and brown remove the top with shears or a knife and throw in the trash.
Manually
Wear gloves and goggles when removing a cactus by hand. Dig deeply around the base of the cactus. From one side, sever any deep roots beneath the plant, ease it out with a rotating motion and throw it in the trash. Be sure to dig up the remaining roots as these can continue to grow. If the cactus is particularly large, cut into portions with a chainsaw. Avoid crushing the plant, or leaving any cuttings as this can lead to further cactus growth.
Manure
After removing the cactus by hand, or cutting into portions with a chainsaw, place the cuttings into small piles and cover with a foot of horse manure. This will cause the cuttings to cease growing, and begin to rot within six months. This can then be disposed of, or used as a fertilizer for the rest of your garden. Return to the roots and remove them using a shovel. Ensure all of the roots are removed otherwise they will continue to grow. Dispose of them in the trash or cover with manure.
Over-watering
All plants need water to flourish and survive, but too much is destructive. When a cactus is over watered, it will begin to grow brown and soft. As the stem begins to rot, the cactus may fall over. If this happens, dispose of the cactus in the trash, either whole or in pieces. Be sure not to leave any pieces of cactus behind, as they may take root. Continue watering the remaining roots there's no further growth.