Question: I am traveling from the US to Australia. What type of under the seat bag is practical, and what should be stored in it for the flight?

Answer: 
The following is a list of the sort of items you should include in your cabin luggage (eg. not your checked luggage):

* All valuables (jewelry, money, etc)
* Important documents (passport, itinerary, hotel reservations, medical prescriptions, etc)
* All electronics (phone, laptop, mp3 players, etc)
* Anything fragile or which might break due to rough or careless handling
* Medication, especially any that you may need to take during the flight such as lactose pills, air sickness pills, pain relief for headaches, etc.
* Anything you may want to use during the flight

What goes under the seat should be the stuff you intend to use during the flight. This may include: some electronics, like a laptop or iPod; a travel pillow; books or other reading material; a blanket or jacket; essential toiletries (eg feminine hygiene products), and so on. 

It's a matter of preference, of course, what you do on the flight. But if you think you'll want to listen to your iPod, it should go under the seat rather than in the overhead so you're not needing to go rummaging. (Going into the overhead on an overnight flight is especially difficult since the crew tends to dim or turn off the cabin lights, so you'll be doing said rummaging in the dark.)

What type of bag you use depends on what needs to go into it. If you're just taking your passport and an iPod and planning to nap most of the trip, you can get away with using a purse. If you're taking more, a backpack may be more practical. The type of bag also depends on what you're actually planning on using in your destination, since taking excessive luggage just for the plane trip isn't always practical.

But in general, the bag should be soft so that it can be fit under the seat and potentially squished a bit. It shouldn't be excessively large in any dimension -- check with your airline for dimensions. And it should be relatively easy to get stuff in or out of it. (Eg a zipper rather than a system of six buckles.)
To add to all the great answers given;

* It can be very cold in the airplane, I like to take a pair of warm socks with me.
* The air can be very dry and rough on the skin, I prefer to take a moisturising cream and lipbalm with me, so I don't start my holiday with itchy, cracked skin and lips.