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: 
Anything valuable and *all* important documents for your travel should be as close to you as possible. This means your passport, boarding passes (unless using electronic ones but I will bring a printed copy too), address at your destination plus any other necessary documentation to pass immigration and customs at your destination airport, including potential layovers and terminal changes which can force you to go through immigration in a country which is neither your departure nor final destination.

Other answers cover some details plus also add things you want during the flight such as snacks and entertainment, space permitting. Make sure you have any prescription medications with you, and going to some destinations, you may need a copy of a prescription or ordnance for the medication. Should there be some delay, this and any additional carry-on, may be your only items for a while. Since it can happen to have unscheduled over-night stops, due to bad weather or missed connections or airplane mechanical issues, I also recommend a change of clothes, at least socks and underwear, plus a few toiletries (toothbrush, toothpaste, deodorant and wipes at least).

As for the bag, Most airlines allow a carry one and a personal item. The personal item has a smaller size limit, so that is the one that I place under the seat. The carry-on goes in the overhead bin where it will be slightly less accessible but arrives most of the time intact at your destination. It happens though that due to limited space, the carry on gets checked at the gate or even to a faraway bin. That's why I always keep the most valuable and essential under the seat in a *personal item* sized-bag.

Most time I use quick access shoulder bag but lightweight backpack can do. A number of these have a dedicated padded compartment for a laptop, ultra-book or tablet which count as your valuables if you travel with such thing.
Under the seat? Hopefully nothing. That's where I store my feet. There's not enough space for a bag.