As a native English speaker, and as far as I know, the only woman in the house, it seemed like time to chime in here. A baby shower happens *before* the baby is born; it is not the same as announcing your new arrival.
It was originally a women-only gathering, and only before the birth of the first child, to help a new mother-to-be gather the basic baby supplies and advice she might need as she enters motherhood for the first time. Nowadays, in Canada and the U.S. (I can't speak for other English-speaking countries), there are (1) sometimes co-ed (both genders) baby showers, or different showers for family, friends, coworkers, and so on, and (2) sometimes baby showers even for the second or third baby.
As far as I can tell, the aim of baby showers is to score as much free stuff as possible, but as James mentioned we don't have kids so I may be biased.
It sounds to me as though Auri is indeed looking for the term "birth announcement". Or, if he's looking for what to write on the card, it's fairly common to write something like "Celebrating Our New Arrival" or "Introducing [insert baby name here]".
Hope that's of some help--or at least offers a few seconds of pointless entertainment.
