My cousin has just had a baby (well, his wife has). I know that the baby is a "second cousin" and a "cousin once removed" - one to me and one to my nephew, but can't remember which way round it is... Anyone?
'Cousin' is if you share the same grandparent, 'second cousin' is if you only share the same great-grandparent.
~ cousin once removed is your ~ cousin's child, ~ cousin twice removed is your ~ cousin's grand child, etc.
so it sounds like the baby is your first cousin once removed, assuming you share grandparents with the baby's parents (oh and it doesn't matter which of the couple actually had the baby, it's still classed as 'their' offspring) http://en.wikipedia.org/wiki/Cousin