The aura-equipment will be attached only to the elf, and the aura's effects and the equipment's effects will apply only to the newly-equipped creature (the elf).
If you instead had an "enchant land" aura attached to a land, you'd lose it as soon as you got the "combo" on the battlefield, before getting a chance to activate the equip ability. It'd be illegal for the equipment to be attached to it, so it'd become unattached, and then since it's also an aura that's not attached to anything, it'd be put into your graveyard as a state-based action.
If on the other hand it was attached to a Dryad Arbor (a land creature) it'd be fine - but if you tried to move it to a regular creature, it wouldn't move, since that's an illegal target for the aura to be attached to.
Vilmar's answer quotes the Bludgeon Brawl rulings which pretty much cover all of this. As for the detailed rules...
303.4b The object or player an Aura is attached to is called enchanted. The Aura is attached to, or “enchants,” that object or player.
This means the aura's effects will apply to whatever it's attached to. There's no notion of "attached-as-an-equipment" versus "attached-as-an-aura". Objects are simply attached.
702.6a Equip is an activated ability of Equipment cards. “Equip [cost]” means “[Cost]: Attach this permanent to target creature you control. Activate this ability only any time you could cast a sorcery.”
So when you activate the equip ability, you attach.
701.3a To attach an Aura, Equipment, or Fortification to an object means to take it from where it currently is and put it onto that object. If something is attached to a permanent on the battlefield, it’s customary to place it so that it’s physically touching the permanent. An Aura, Equipment, or Fortification can’t be attached to an object it couldn’t enchant, equip, or fortify, respectively.
It's not totally explicit, but this is the closest I could find to something saying that an object can only be attached to one object, i.e. if A is attached to B and you then attach it to C, it's no longer attached to B. (Clearly this is how equipment works, so I don't think we need worry about it being wrong.)
Taken together, this means that when you equip your aura-equipment to a new (legal) object, it becomes attached only to that object, and the aura's effects apply only to that new object. In your case, this means that the elf gets +X/+0 and flying, and the bear doesn't get anything.
As for the case where you try to attach to an illegal target, it's pretty simple:
701.3b If an effect tries to attach an Aura, Equipment, or Fortification to an object it can’t be attached to, the Aura, Equipment, or Fortification doesn’t move. If an effect tries to attach an Aura, Equipment, or Fortification to the object it’s already attached to, the effect does nothing. If an effect tries to attach an object that isn’t an Aura, Equipment, or Fortification to another object or player, the effect does nothing and the first object doesn’t move.