In general, search on AA.com for award availability. BA
If you want to book a flight with Avios, it is known from Dan's Knowledgebase
that BA.com doesn't show availability very well, but some flights definitely are searchable there. Nonetheless, it's easier to just search AA.com, and if you find a MileSAAver availability, check BA.com to see if it shows up there, too. If it does, book it there. If it doesn't, you can always call up BA and reference the flight you found. Note that there is usually a very long hold time before BA picks up, and they will charge a phone booking fee. If the flight was unsearchable on BA.com, try fighting that fee, but YMMV.IB
IB availability will not show up on AA.com You can call AA to find IB availability, but it's hard without being able to browse dates and have to ask the rep to check flights over the phone. Searching on Qantas.com will show IB flights - supposedly if they appear there, they can then be booked by calling AA.
OneWorld Explorer [OWE] The OWE has officially been cancelled [4/8/14] as part of the latest AA no-warning changing to its award program. Here's what the OWE looked like:
This is an around-the-world [RTW] award that is calculated not by the mileage cost of specific flights, but rather by the actual distance travelled on your trip. It is especially valuable for flying business class, as the difference in miles needed between coach and business is not very great on longer trips. See this chart to see the different tiers of how many miles it costs to travel different distances. You can take up to 16 segments on a trip, but there are MANY rules to follow when booking such an award.
Be prepared to spend a lot of time when planning this trip. (Personally [@Yehuda], it took 2 months with a few weeks of break in the middle from frustration.) Availability changes everyday, so you need to constantly check back to see if flights you needed are now available. It will also likely require changing your itinerary/order of stops numerous times. An itinerary can be held for 5 days (and can be extended past that on Day 5, if they are willing to rebook each flight for you, getting you another 5 days), but itineraries with IB flights can usually only be held for 24 hours. You will find that you will have to use hub airports since you're limited to OW partners. You will not find a flight from CDG-FCO without either stopping off in LON or MAD - both of which are out of the way. It's unfortunate, as it adds unnecessary distance miles to your trip, but it can be done. Remember that BA flights have high YQ, IB/AB have semi-high YQ, US/AA have very little YQ. (Unsure about other airlines, as my trip was TLV-Europe-NYC, so those were the only airline I had to work with.) You should check flight availability, again, on AA.com but the ticket must be booked on the phone (and they will charge a $25 phone booking fee, but again, try to fight it since this ticket cannot be booked online - although, honestly, by the time you've ticketed, they will likely have helped you out with much more than $25 worth of time). Technically, you can book it with any OW partner, but the odds are you probably only have enough AA miles to book this trip. AA reps definitely know about this award, although most have never booked it before. You will find that after calling so many times, you'll know more about the rules than they do. Once you have your perfect itinerary, they will have to send it to their rate desk to price it out. This can take ~24 hours. Because it takes time, if you have IB flights on the itinerary, they will make sure IB allows the flights to be held for longer than 24 hours, but confirm this with the rep before you hang up. Once it's priced out, you can pay for it. You can use 2 or more AA accounts to pay for it, but they will require each account to have enough miles to pay for 1 ticket. (e.g. If it costs 80K pax and you have 2 pax, they won't ask you to have 1 account with 160K in it, but they will require 2 accounts with at least 80K in each.)
Feel free to contact @Yehuda with any questions regarding the OWE.