Grandparents / grandchildren
The parents of your parents are your grandparents – grandmother and grandfather. You are their grandchildren – either a granddaughter or a grandson. If your grandparent has a sister, she is your great-aunt. If your grandparent has a brother, he is your great-uncle. (And you are either his or her great-niece or great-nephew.)
The mother of your grandmother or grandfather is your great-grandmother. The father is your great-grandfather. If you go back another generation, the grandmother of your grandmother / grandfather is your great-great-grandmother. The grandfather of your grandparent becomes your great-great-grandfather.
Grandparents / grandchildren
The parents of your parents are your grandparents – grandmother and grandfather. You are their grandchildren – either a granddaughter or a grandson. If your grandparent has a sister, she is your great-aunt. If your grandparent has a brother, he is your great-uncle. (And you are either his or her great-niece or great-nephew.)
The mother of your grandmother or grandfather is your great-grandmother. The father is your great-grandfather. If you go back another generation, the grandmother of your grandmother / grandfather is your great-great-grandmother. The grandfather of your grandparent becomes your great-great-grandfather.
Second families
If your mother or father remarries, you can acquire a new family and set of relatives. For example, if your father marries a second wife, she becomes your step-mother. Any children she already has become your step-sisters or step-brothers.
If your mother or father remarries and has children, they become your half-brothers or half-sisters.
You might also hear people talking about their biological brother / sister etc, to mean a brother who is related by blood, rather than by marriage.
Types of family
nuclear family = mother, father and children: "The traditional British family unit is a nuclear family."
single-parent / one-parent family = a family which only has one parent (because the parents are divorced, or because one of the parents has died): "There are more and more single-parent families in the UK."
immediate family = your closest relatives: "Only immediate family members attended the funeral."
extended family = your entire family: "The wedding invitations were sent to the entire extended family."
close-knit family = a family where the members have close relationships with each other: "They are a close-knit family."
dysfunctional family = a family where the members have serious problems with each other: "He comes from a rather dysfunctional family."
blood relative = a relative connected to you by "blood" rather than through marriage: "She's not a blood relative, but we're still very close."
Expressions with family
family gathering = a meeting / celebration of family members: "There's a small family gathering next week."
family resemblance = where members of the family look / act similar: "You can see a distinct family resemblance between the father and the son."
to start a family = to start having children: "They want to wait a couple of years before starting a family."
to run in the family = a characteristic that is common among family members: "Baldness runs in his family."
to bring up / raise a family = to have and look after children: "It's difficult to raise a family on one income."
a family car = a car big enough to transport a family: "The Volvo Estate is a popular family car."
family-size = large quantity item: "We need to buy family-size packets of biscuits!"
family-friendly = a policy that favours families: "This hotel is family-friendly."
family doctor = a doctor who looks after general medical needs: "There are a number of good family doctors in this area."
family man = a man who prefers to spend his time with his family: "John is a family man."
family values = traditional ideas about what a family should be: "Some political parties often emphasise family values and the importance of marriage."
family name = surname: "What's your family name?"