When the word "removed" is used to describe a relationship, it indicates that the two people are from different generations. You and your first cousins are in the same generation (two generations younger than your grandparents), so the word "removed" is not used to describe your relationship.
The words "once removed" mean that there is a difference of one generation. For example, your mother's first cousin is your first cousin, once removed. This is because your mother's first cousin is one generation younger than your grandparents and you are two generations younger than your grandparents. This one-generation difference equals "once removed".
Twice removed means that there is a two-generation difference. You are two generations younger than a first cousin of your grandmother, so you and your grandmother's first cousin are first cousins, twice removed.
Your mother's first cousin is always your first cousin, once removed. That is at the core of what first cousins and removed cousins are. That relationship will never change; your mother's first cousin will always be your first cousin, once removed. The term "cousin by marriage" is relatively new and undefined, and for many it really does not mean anything. Some people use the term to refer to someone who has married a cousin. In that sense, if someone marries your "first cousin once removed," that person would be your "first cousin once removed by marriage." It is completely incorrect to claim that someone who marries your first cousin is your "first cousin once removed" simply because of that marriage.
It means they are one generation removed from you on your family tree. Your second cousin once removed is either your parent's second cousin, or your second cousin's child.
Your wife's first cousin, once removed. The English language has no shorter term or special name for that person.
first cousin once removed
first cousin once removed
Your father's first cousins are your first cousins, once removed.
The expression is akin to once bitten twice shy.
Your first cousin's daughter is your first cousin, once removed.
hypophysectomized - having the pituitary gland removed by surgery
In designating the relationship of cousins, removed means that one cousin is one or more generations further removed from the common ancestor than is the other cousin.
Cousin once removed means that the cousin relationship is removed by one generation. Example: Your dad's first cousin would be your first cousin once removed because you are from one generation later than your dad. Your first cousin once removed would be your childs's first cousin twice removed because your child is two generations later than your dad, etc.
Your grandfather's niece is the first cousin to one of your parents. That would make you first cousins once removed. Most families just use the term cousin.