The term 'First Death' usually refers to the literal first death of an immortal which triggered their latent immortality. These deaths are, as far as is known, usually violent, and always unnatural. Both men bearing the title Highlander, Connor and Duncan MacLeod, met their first deaths in clan battles.
In Highlander: The Raven, Amanda's friend and protege Nick Wolfe - a mortal with the potential to become immortal - was dying slowly from poison. Certain that Nick would die permanently if the poison was allowed to run its course, she chose to shoot him in order to trigger his immortality and thus save his life.
In Highlander: Endgame, Duncan MacLeod killed his wife, Kate Devaney, in order to trigger her immortality and avoid the heartbreak Connor MacLeod and Heather MacDonald endured as a couple, when Heather aged and died naturally as a mortal. Connor noted that he did not envy Duncan's position when the younger MacLeod asked him for advice on whether or not he should take action regarding Kate's immortality. Kate, however, was both horrified and incensed by Duncan's murder of her. She later joined Jacob Kell's Gang working against the MacLeod's under the pseudonym of Faith.
In the audio series story, The Promise, Methos snidely refers to the almost inevitable discussion of one's first death as the "standard immortal icebreaker" and finds the topic boring. He is, of course, too old to remember his own first death; his memories only go as far back as taking his first head and Quickening, about 5,000 years. His first death occurred an unspecified length of time prior.