Euthanasia (AKA mercy killing) is the deliberate killing of another person with the motive of ending their suffering. This can be achieved by doing something (eg. a lethal injection) or by failing to do something (eg. withholding life-saving treatment) in order to cause or hasten death.
It is morally equivalent to assisted suicide which is where someone who knows what is about to happen allows someone to kill themselves or provides the means for them to do it.