You can use Lady to save the card from various effects that would cause it to leave play, including being killed, discarded from play, returned to your hand, removed from the game, placed on top of your deck, etc. You cannot save the card from being sacrificed, from terminal burn, or from any effect that says "cannot be saved". You cannot save the card if it would leave play as part of the cost of initiating an ability (for instance, Grey Wind (FtR)).
You may include both the character and the attachment version of Lady in your deck, but you can still only have a total of 3 cards with the title "Lady". If you have Lady the character in play, Lady the attachment can enter play as a duplicate, and vice versa. If the character is in your dead pile, the attachment cannot enter play.