Over the years, I have performed at numerous "parties". From birthdays and wedding anniversaries through to military Mess functions and christenings, I don't think there's been a "type" of event that I haven't played at. In the unlikely event that you are doing something completely different, I am only too happy to discuss what you have in mind.
There really isn't a set format for these as each one has it's own purpose. At birthday parties for instance, I have been asked to play for the grand entrance of the birthday cake. The tune "Happy Birthday" is within the musical scale of the bagpipe and is often well received.