One’s first cousin once removed is the person’s mother’s cousin. When two people are related once removed, they are connected but from different generations. One’s first cousin once removed is also one’s cousin’s child. This and the phrase “second cousins” are commonly mistaken.
Children of one’s parents’ cousins are considered second cousins. Second cousins have a common great-grandparent at a minimum. The great-grandparent of the younger generation is the grandparent of the elder generation when they are first cousins once removed.
Double first cousins are a somewhat typical connection in which two brothers marry two sisters and have offspring. The kids share both sets of grandparents and are first cousins twice over.