Euthanasia is an intervention undertaken with the intention of ending a life to relieve suffering, for example a lethal injection administered by a doctor
Assisted suicide is any act that intentionally helps another person kill themselves, for example by providing them with the means to do so, most commonly by prescribing a lethal medication
Assisted dying is usually used in the US and the UK to mean assisted suicide for the terminally ill only, as for example in the Assisted Dying Bills recently debated in the UK
http://www.bbc.com/news/world-europe-37395286