Your first cousin's grandkids would be your first cousins twice removed. Once removed means they are one generation removed from you (i.e. your first cousin's child). Twice removed means two generations removed from you (i.e. your first cousin's child's child).
As for second/third, think of it like this: numbered cousins are essentially the same generation as you, so they'd be roughly the same age as you. Your first cousin and you share the same grandparents (they are your parent's sibling's child). Your second cousin shares the same great-grandparents (they are your grandparent's sibling's grandchild). Your third cousin shares the same great-great grandparents (they are your great-grandparent's sibling's great-grandchild).
As the numbers get higher, the relation gets more distant.
