Here are the largest airlines in some different categories. United, as the leader in Revenue Passenger Miles (RPMs), is generally accepted as the largest.
RPMs: United, then American
Revenue: United, then American
Profit: American, then Delta
Passengers: Delta, then United
Fleet: American (648), then Delta (583)
As you can see, the same three airlines show up again and again, so it's pretty clear that by any measure these are the three largest.
As for foreign airlines (non-US):
RPMs: British Airways, the JAL
Revenue: British Airways, then Lufthansa
Profit: British Airways, then Lufthansa
Fleet: British Airways, then Lufthansa