It's a sad thing when a beloved fish friend comes to the end of its lifespan, when it won't eat or swim, or when it has succumbed to serious injury or disease. It can be difficult to take the responsibility for ending the life of another being but many aquarists feel it's worse to allow a creature you know to be in pain to suffer, when its ailments cannot be cured.
Older methods include flushing, fast freezing, fast boiling, decapitation, and so on, but are now considered inhumane. Modern fish-keeping methods have suggested a more humane method which is simple, clean, and even inexpensive.
Clove oil (also known as eugenol) is sold in most drugstores as a toothache remedy, and has the remarkable quality of putting fish to sleep. Using clove oil to anaesthetize your ailing fish first will make almost any method of euthanasia more humane, but the quickest and cleanest way is to add a clear grain alcohol like vodka to the water inhaled by the sleeping fish.
For small fish, under three inches long:
1) Fill a measuring cup with tank water, noting the volume, and pour it into a separate container (opaque sides will minimize fish stress). Place the fish in the container.
2) Fill a small jar ? full with water from your tank, add one drop of clove oil, put the lid on, and shake thoroughly until the water turns a cloudy white.
3) Pour about a quarter of the white liquid into your container, and allow it to sit for about ten minutes. The fish will begin listing and then sink to the bottom, but the gills will still move every few seconds. If the fish is still moving after ten minutes, shake up the jar again and add another quarter of its contents to the container.
4) When the fish has stopped moving completely except for the gills, add vodka or a similar clear grain alcohol to the water. You'll need to use about a quarter of the original container's volume in vodka: if you put 12 ounces of water in the container, you'll need 3 ounces of vodka (25% of 12 oz) to make the process quick and effective.
5) Check the fish after twenty minutes; if there is no gill movement after sixty seconds, the process is complete; if you do see any movement, add another dose of clove oil, and then another dose of vodka.
For larger fish you'll need proportionally more clove oil: about 10 drops per gallon of water in the container; and you may need to wait longer before it takes effect. Once the fish is asleep, use the same 25% dose of alcohol to ease them on their way.
An aquarist hopes that they will never have to perform this duty. I think it's better to know that there's a humane way, and never need the information, than to need the information and do something cruel to your fish because you didn't know.