Jesus, like everyone else, also had uncles, aunts and cousins.
Elizabeth was Mary's cousin, and she was married to Zechariah.
The Gospel of Luke presents them as two very old people, who always wished and asked God for a child. One day, an angel appeared to Zechariah in the temple and told him that their wish was fulfilled: Elizabeth stopped being sterile and gave birth to John the Baptist.

John the Baptist is the son of Elizabeth and Zechariah. He is Jesus' cousin, and was born six months before him. He is the only holy person, alongside Jesus and Mary, whose earthly birth and death—ordered by King Herod—are celebrated.
The fact that he is called the Baptist already indicates what he was doing: in the River Jordan, he baptized sinners and called everyone to prepare for the coming of the Messiah. He also baptized Jesus, and he told his friends to follow him because he was the one they were waiting for.

We do not know anything about Jesus' Grandparents. The name of St. Joseph's father was Eli or Jacob. Mary's parents, according to some legends, were Joachim and Anna.
These same legendary texts, called apocryphal gospels (because they are not in the Bible and historically not very sure), say that the couple had no children and were getting on in years. Yet, one day, while Joachim was working in the fields, an angel appeared to him to announce the birth of a child, and Anna too had the same vision. They called their daughter Mary—that means, "Loved by God."