An Aura spell on the stack targets the object or player which it will enchant upon entering. Thus, an Aura spell is a "spell with a single target", which means you may use Swerve to change that target. Doing so will cause it to enter enchanting the new target rather than the original one.

Swerve targets only the spell whose target will be changed. It doesn't directly affect the original target of that spell or the new target of that spell.

You don't choose the new target for the spell until Swerve resolves. You must change the target if possible. However, you can't change the target to an illegal target. If there are no legal targets, the target isn't changed. It doesn't matter if the original target of that spell has somehow become illegal.

If you cast Swerve on a spell that targets a spell on the stack (like Cancel does, for example), you can't change that spell's target to itself. You can, however, change that spell's target to Swerve. If you do, that spell won't resolve when it tries to resolve because Swerve will have left the stack by then.

If a spell targets multiple things, you can't target it with Swerve, even if all but one of those targets has become illegal.

If a spell targets the same player or object multiple times, you can't target it with Swerve.