The Regenerate ability adds the following (paraphrased) to a given permanent:
"The next time this permanent would be destroyed (either by lethal damage or by a 'destroy target X' spell), prevent that destruction. Tap that permanent and, if it is a creature in combat, remove it from combat."
Regeneration cannot be used to call something back from the graveyard: it can only be used to save something from being sent to the graveyard either by a direct destruction spell/ability or lethal damage (both of which are called "destruction events"). Once the card is in the graveyard, you'll have to use some other ability that specifically says to return the card to play or to your hand, such as a creature with the Unearth ability.
The easiest way to think about it, as most players do, is that regeneration adds a shield around the permanent. The next time the permanent would be destroyed, the shield goes off and brings it back to life, with full health and all.
To use Broken Fall, it must already be in play. Since it is an enchantment, you can only put it out during your own turn and only at a time a sorcery could be played. That means you can use its ability in response to one destruction event, but you can't play it fast enough to respond to one if the card is still in your hand.
So, if the destruction spell or lethal damage is being sent to two or more of your creatures, you won't be able to play Broken Fall again before that effect resolves. In that case, you have the choice of saving only one creature if Broken Fall was already on the field.
For more info on this, look up regeneration and the timing of spells in the comprehensive rules, linked below.