Gaffer Gamgee was a common nickname for the father of Samwise Gamgee, who was more properly called 'Hamfast'. He presumably acquired the name 'Gaffer' later in his life, because it implies a person of advanced years. This was certainly true of Hamfast, who was old enough to remember Bilbo's return from Erebor, and was no less than ninety-two years old when his son Sam left the Shire with Frodo.
'Gaffer' has another meaning, too, which is no doubt also relevant. In British slang, a 'gaffer' is a foreman, a person who supervises a job. At the time of Bilbo's Birthday Party, we're told that Hamfast, who was then seventy-five, oversaw the gardening at Bag End, but was helped by his son Sam. In that sense, then, Gaffer Gamgee actually was a 'gaffer'.