Cousin (a.k.a "first cousin")
Your first cousins are the people in your family who have two of the same grandparents as you. In other words, they are the children of your aunts and uncles.
Second Cousin
Your second cousins are the people in your family who have the same great-grandparents as you., but not the same grandparents.
Third, Fourth, and Fifth Cousins
Your third cousins have the same great-great-grandparents, fourth cousins have the same great-great-great-grandparents, and so on.
Removed
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.