Latin phrase meaning 'on the point of death' (literally 'in the article of death'). Why article? My guess is that because that literally means 'joint' (as in articulation) it was thought of as the connection betwen this world and the next.
Use in medical senses to refer to the interesting thing that happen when someone is dying. It is not an instantaneous thing: someone in articulo mortis may be aware that they are dying, and perform actions and say things.
This brings in the legal sense. Certain things are special if done in articulo mortis. I am not a lawyer and the following are just general descriptions of the kind of things that are privileged by the circumstance. Some of this may be traditional English law which no longer applies in this or that jurisdiction.
When the dying person tells someone how they died (e.g. describes a murderer or assailant), that witness's evidence of this is valid, i.e. it is not disqualified as hearsay.
Deathbed confessions are in general treated as true. This, like the above, is from a residual religious sense that the person is about to meet their Maker, and so have no ulterior worldly motives for making their statements. These declarations are treated as if on oath, even though in the circumstances it is not possible to administer a legal oath and do other legal things such as cross-examination.
Gifts may be made in articulo mortis. The person's heirs (by blood or by will) are not entitled to the things given away. But it's been held that this doesn't extend to things that can't normally be simply given away, such as houses and cars: a proper transfer of deeds still has to be made for those.
A will may be made by declaration. This is called a nuncupative will. It may be valid if declared in articulo mortis even if wills must otherwise be in writing.
Related to this is the privileged will, made by someone who is professionally in a situation of danger of death - seamen at sea and soldiers on active service. However, this kind of will can be made calmly and when the person in danger is not actually dying.