This is most often done to patients who are in a Persistent Vegetative State and will probably never recover consciousness and to disabled babies or people with severe brain damage. .
The reasons that push people to end their life vary, one of the most common reasons is to end the pain of the person(physical or psychological), but who said that the only way to releif the pain is death? And if so then what were painkillers made for? Some terminally ill patients end their life to save the money their family spends on them! This is not the smartest of reasons to make a person end his life because naturally the family would rather have the person back with them well than having money. Another common reason is to guarantee a "dignified death" (whatever that means). They think that being very ill and not being able to do everything they used to do while they were well and losing their independence is an unacceptable loss of dignity. Well, even if it was they also need to know that god washes away the sins of the people who suffer while they are ill. Some tend to end their own lives because they know for sure that they"re dying anyway and they want to have control over the process! Needless to say this reflects great disrespect to god and religion, it's like saying "you can't fire me, I QUIT". Those reasons among others are not enough justification for ending one's life.
Needless to say, Euthanasia does not sound like the right thing to do because the facts are so clear and any sane person would know what is right and what is wrong and naturally ending own life is non ethical and against all religions. Killing yourself is not a crime in most parts of the world because when u commit suicide u die so it will be hard to punish a dead person. But KILLING a person is a different story (you might have noticed that I have underlined the word "killing" in the definition of euthanasia) killing is illegal all over the world and helping a person die is a way of killing a person.