An obituary was a written passage about the life of an individual which was published immediately following their death. In the 20th century, they were typically published in the obituary column of newspapers.
Spock viewed the obituary of Edith Keeler in his tricorder readings of the Guardian of Forever while in the 1930s. This helped him realize that her death was a necessary part of history. (TOS: "The City on the Edge of Forever")
In 2151, Trip Tucker characterized Malcolm Reed's despair over being stranded by saying he had been writing his own obituary since they saw what they believed to be the destruction of Enterprise NX-01. (ENT: "Shuttlepod One")
Obituaries were normally written by a third party or relative of the deceased, but Professor Gideon Seyetik took the unusual step of writing his own obituary before his death in 2370. He asked Benjamin Sisko to send it to the Daystrom Institute on his behalf to be published. (DS9: "Second Sight")