The children of a first cousin would be called a first cousin, once removed. First cousins are defined as children of siblings.
"Once removed" always refers to the children of a cousin, whether it be a first or second cousin. A second cousin is defined as the children of the children of siblings. For example, Frank and Johnny are brothers. Frank has one child named Sue and Johnny has one child named Melissa. Melissa and Sue are first cousins. Sue has a child named Bobby and Melissa has a child named Armando. Armando and Bobby are second cousins. Melissa is a first cousin, once removed to Bobby. Sue is a first cousin, once removed to Armando. This pattern can repeat itself through many generations of cousins. "Removed" simply means that two people are cousins with a generation in between them, or a generation removed.
Knowing the way that cousins work and how they are named is a way for people to track their genealogy. A person who understands who their third cousin twice removed is will be able to better understand where they came from. People can also have a better understanding of the family members they share with certain cousins.