Everyone gets invited to the Captain's Welcome Aboard Cocktail Party, which as others have noted is on the first formal night. The ship's orchestra will be playing dance music, waiters will be circulating with trays of assorted drinks and canapes. Just like any cocktail party really - as in being boring too. The Captain or some other senior officer will greet each guest at the door and you can have your photo taken - which you can purchase later. After a while the Cruise Director will take the stage and introduce the Captain. The Captain will then introduce his department heads (Staff Captain, Chief Engineer, Doctor, etc.) and make some brief comments. The band then plays a little more while some guests dance and then you leave.
As you may tell, I am not a big fan of these. However, I usually go to the past passenger party that happens later in the cruise. These are usually better and they typically have prize drawings.