There are two apps I can't stop recommending to people. The first is Square, but I usually reserve that one for people who run small businesses. It's a free and dirt-cheap system for processing credit cards.
Literally. The app provides a shared calendar that's color-coded for each member of your family. Thus, you can see at-a-glance who's going where, and when. (This is especially helpful for parents who need to know, say, what time soccer practice is being held and who's supposed to pick up the kids.)
When you add an appointment, you can designate which family members are involved. The calendar gets synced not only with the other members of your Cozi circle, but also with Cozi on the Web (which is where you can manage settings, add users to your account, and so on).
In addition to the calendar, Cozi provides a to-do list, a shopping list, and even a family journal that can include both text and photos (a great way to chronicle a family vacation).
The ad-supported versions of the app are free (as is the Web-based version of Cozi), or you can pay $5.99 for ad-free Cozi Family Organizer Premium. Best of all, everything works cross-platform: Cozi runs on my iPhone, my wife's Android phone, my daughter's iPod Touch, and so on.
My one and only complain is with the calendar, which offers only a scrolling day-by-day view. It works well enough, but I'd really like some list-view options that would show only upcoming events--and allow me to filter them by person. A month view would be nice, too.
Still, this is without question one of my favorite apps, and I can't recommend it highly enough for busy families.
(Here's another "best app you've never tried":