The Gaffer has been Bilbo's gardener at Bag End for many years; his son, Sam Gamgee, is destined to take over the position when Frodo is Master of Bag End. The Gaffer is a good friend of Bilbo's, and while he is always willing to gossip about Bilbo and the Baggins family down at the inn The Ivy Bush, he nips truly mean-spirited comments in the bud. The Gaffer is the one who tells the story of Frodo's parents' drowning at Brandy Hall to his friends at the inn, but when nasty rumors arise that "[Primula] pushed him in, and [Drogo] pulled her in after" (1.1.16), the Gaffer immediately shuts them down. He's a loyal but conservative Hobbit, filled with old sayings that Sam likes to repeat during his travels with Frodo.
Black Riders ask the Gaffer, among others, where to find Frodo. A loyal friend, the Gaffer tells them only that "Mr. Baggins ha[s] left his own home for good" (1.3.91). Sam passes this information on to Frodo, so that he knows the mysterious Black Riders are looking for him all over the Shire.
One last note about the Gaffer: when Sam looks into the Mirror of Galadriel, he sees a vision of the Gaffer kicked out of his home, as builders rip up the neighborhood of Bagshot Row. Sam almost gives up the quest right then, out of worry for his father, but he continues onward. Still, Sam's worry over the Gaffer indicates how close he is to his father, and how strong his emotional ties remain to the Shire.