There are three main 'alliances': OneWorld (eg Qantas), Star Alliance (eg Air new Zealand) and Sky Team (eg Alitalia). Virgin Australia is not in any alliance, but is partnered with airlines who are in alliances.
The airlines in the same 'alliance' promote that they work together to make flying between airlines easier, although sometimes it seems like a loose collection of warring tribes

.
For instance, if you are QFF, you will earn Qantas points and Status Credits on many flights on other OneWorld Carriers (usually the cheap-o fares don't earn), but it varies how much. But you won't earn on Lufthansa for instance, which is in Star Alliance.
Sometimes there are 'partner' airlines, such as Emirates, West Jet and Alaska, for Qantas which aren't part of OneWorld, where you can also earn QFF points (but not Status credits).
If you have 'status' on a OneWorld airline (eg Gold or Platinum with Qantas), you can access lounges, more checked luggage, priority boarding etc of the other OneWorld airlines, but the rules can be complex, and there are exceptions.
Before they stuffed up the rules about 'interlining' bags, you used to be able to check your bags through connecting flights on other Alliance airlines, but that's now airline dependent.
Alliances are good, but you have to be careful not to organise your travels just to keep within the same alliance, just to earn points etc. You can spend a lot more that way, and the value of points is hard to judge, as with Qantas, for instance, its very hard to book the flights you might want (overseas), especially in premium cabins.