first anniversary of a person's death
Refers specifically to the Buddhist memorial service held on the first anniversary of a death, counting from the date of death. Often used in the context of 一周忌の法要.
See also: 忌日 (きにち)
Relatives gathered for the first anniversary of my grandmother's death.
Next month is the first anniversary of my father's death, so I'm preparing for the memorial service.
三回忌 is the second anniversary memorial service (counting the death as the first year), whereas 一周忌 is the first anniversary.
Compound of 一 (one), 周 (circuit, round), and 忌 (mourning, anniversary of death). Literally 'one-round mourning anniversary'.