[ parents and siblings ]
Sam was a child of Hamfast Gamgee and Bell Goodchild. Not much is known about this couple, other than that Hamfast was gardener in Bag End estate, in which Sam was helping him. Hamfast was called a Gaffer by Sam, which indicates that Hamfast was person of a strong personality.
Sam had two brothers: Hamson and Halfred and three sisters: Daisy, May and Marigold.

[ wife and children ]
After the War of the Ring, Sam married Rose Cotton, with whom he had thirteen children: Frodo Gardner, Elanor, Rose, Merry, Pippin, Goldilocks, Hamfast, Daisy, Primrose, Bilbo, Ruby, Robin, and Tolman (Tom). Samwise always felt affinity to Rosie, they were close friends most of childhood, and the girl expected him to propose, yet Sam at that time moved with Frodo on their perilious mission.
Rose and Sam were married in 3020, in May. From then, and since Frodo left to Undying Lands, the couple settled in Bag End.
Mr. Tolkien said that normal life led by Rosie and Sam was needed, after all the pain, adventure and danger. Sam was a hero, but heroes also need normalcy. Sam, marrying to Rose, huffed in arms of someone who loved him and who grounded him on the earth he walked on.

[ best friend ]
But... but! Speaking of family, and even if I plan to add special section to this topic, I can't not mention Frodo! Friends, especially best ones, for whom you would give your own life and move mountains, also are family, in non literal meaning of this word. Frodo was Sam's family by heart, not by blood, or marriage, he was first and last person in Sam's life of such unique importance.