Generally speaking a cousin is a relative who has descended from a common ancestor. However, when we use the term cousin we are usually referring to a child of an uncle or aunt. More accurately we are referring to a first cousin. Thus, a first cousin is the term used to describe the relationship between the children of siblings (brothers/sisters). Using the same terminology, a second cousin is the term used to describe the relationship between the children of first cousins. Similarly, a third cousin is the term used to describe the relationship between the children of second cousins and so on. First, second and third cousins refer to the relationship between the descendants of a common ancestor at the same generation.
A cousin a number of times removed is a term used to describe a relationship between cousins when they are not of the same generation. For example, a first cousin once removed is the term used to describe the relationship between a person and their first cousin's child. It also describes the reciprocal relationship, i.e. the relationship between a person and their parent's first cousin. Similarly, a second cousin once removed is the term used to describe the relationship between a person and their second cousin's child. A first cousin twice removed is the term used to describe the relationship between a person and their first cousin's grandchild and also the relationship between a person and their grandparent's first cousin. Thus, once removed refers to a difference of one generation and twice removed refers to a difference of two generations.
Hopefully, the above definitions are made clearer by the tree diagram below. The different colours denote descendants at the same generation.