These are called "split cards". They were introduced in Invasion block and then returned during Ravnica and Time Spiral blocks.
When you cast one, you pick one side of the card to cast and then treat the other like it doesn't exist. The whole card will go in your graveyard. There's no way to cast both sides at once (unlike Entwine spells).
As far as what happens in other zones, the basic idea is that the card has two values. For instance, Hit / Run has a combined mana cost of "3 and 5" and its color are red, green, and black.
As far as what "3 and 5" really means in all sorts of situations, honestly it's probably easiest to quote the whole comp rules:
708. Split Cards
708.1. Split cards have two card faces on a single card. The back of a split card is the normal Magic
708.2. In every zone except the stack, split cards have two sets of characteristics and two converted
mana costs. As long as a split card is a spell on the stack, only the characteristics of the half being
cast exist. The other half’s characteristics are treated as though they didn’t exist.
708.2a If a player casts a split card, that player chooses which half of that split card he or she is
casting before putting it onto the stack. Only the half that is being cast is considered to be put
onto the stack.
708.3. Each split card that consists of two halves with different colored mana symbols in their mana
costs is a multicolored card while it’s not a spell on the stack. While it’s a spell on the stack, it’s
only the color or colors of the half being cast.
708.4. Although split cards have two castable halves, each split card is only one card. For example, a
player who has drawn or discarded a split card has drawn or discarded one card, not two.
708.5. An effect that asks for a particular characteristic of a split card while it’s in a zone other than the
stack gets two answers (one for each of the split card’s two halves).
- Example: Infernal Genesis has an ability that reads, “At the beginning of each player’s
upkeep, that player puts the top card from his or her library into his or her graveyard. He or
she then puts X 1/1 black Minion creature tokens onto the battlefield, where X is that card’s
converted mana cost.” If the top card of your library is Assault/Battery when this ability
resolves, the game sees its converted mana cost as “1, and 4.” You get five creature tokens.
708.6. Some effects perform comparisons involving characteristics of one or more split cards in a zone other than the stack.
708.6a An effect that performs a positive comparison (such as asking if a card is red) or a relative
comparison (such as asking if a card’s converted mana cost is less than 2) involving
characteristics of one or more split cards in any zone other than the stack gets only one answer.
This answer is “yes” if either side of each split card in the comparison would return a “yes”
answer if compared individually.
708.6b An effect that performs a negative comparison (such as asking if cards have different names)
involving characteristics of one or more split cards in any zone other than the stack also gets
only one answer. This answer is “yes” if performing the analogous positive comparison would
return a “no” answer.
708.6c If an effect performs a comparison involving multiple characteristics of one or more split
cards in any zone other than the stack, each characteristic is compared separately. If each of the
individual comparisons would return a “yes” answer, the whole comparison returns a “yes”
- Example: Void reads, “Choose a number. Destroy all artifacts and creatures with
converted mana cost equal to that number. Then target player reveals his or her hand
and discards all nonland cards with converted mana cost equal to the number.” If a
player casts Void and chooses 1, his or her opponent would discard Assault/Battery
because the game sees its converted mana cost as “1, and 4.” The same is true if the
player chooses 4. If the player chooses 5, however, Assault/Battery would be unaffected.
708.7. If an effect instructs a player to name a card and the player wants to name a split card, the player must name both halves of the
split card. An object has the chosen name if it has at least one of
the two names chosen this way.
Thus, we get these interactions:
If your Dark Confidant digs up Hit / Run, he'll see "3 and 5" (thanks to rule 708.5) and you lose 8 life. Brutal!
Bloodbraid Elf's cascade ability, on the other hand, is a positive characteristic comparison, so that "3 and 5" allows Hit / Run to satisfy the condition if either value would individually (see rule 708.6a). That means that, when Bloodbraid Elf finds Hit / Run, the entire card Hit / Run satisfies the condition because Hit would. When you actually go to cast Hit / Run without paying its mana cost, you go still through the normal spell casting procedure, including choosing which side you're casting. That means you can go ahead and cast Run, if you want!