Uncle and aunt - your parents’ brothers and sisters.
Niece and nephew - son and daughter of your brothers and sisters.
First cousin (aka cousin, full cousin) - children of your parents’ brothers and sisters. You and your first cousins have one set of grandparents in common.
Double first cousins (aka double cousins) - if a pair of brothers marries a pair of sisters, their children are not only first cousins, they’re also double first cousins as they have both sets of grandparents in common.
Second cousin - you and the children of your parents’ first cousins are second cousins. You share at least one common great-grandparent. Your child and your first cousin’s child are second cousins.
Third cousin - you and the children of your parents’ second cousins, are third cousins and you share at least one great-great-grandparent. And so on for fourth, sixth, etc...
First cousin once removed - a relationship that is removed, is one that exists in two different genealogical generations. Your father and his brother may have been born 20 years apart, but they are still of the same generation. Your parents’ first cousin, is your first cousin once removed. The child of your first cousin is also your first cousin once removed - your grandparent is that child’s great-grandparent. You also carry on with second cousin once removed, third cousin once removed, etc...
Grandaunt and granduncle - the sisters and brothers of your grandparents.
Great-grandaunts and great-granduncles - your great-grandparents’ sisters and brothers.
In-laws - family by marriage. Your spouse’s parents, spouses of your siblings and spouses of your spouse’s siblings. Your sister’s husband is an in-law, but none of his siblings are. Your husband’s sister’s husband is your in-law, but none of his brothers are.
Affinity relatives - your spouse’s blood relatives, the in-laws that are biologically related to your spouse.