I've watched so many different videos and seen a lot of different facts about the Rune system in Diablo 3. It all keeps changing. I'm not really sure which is the correct answer. I'm hoping someone can clarify how this system works and what these runes do.
As of a recent beta patch, runes are no longer items that are dropped. Instead they are basically skill modifiers that are unlocked as you level up. Each time you level up, you will unlock new skills and/or rune choices that affect your existing skills. You choose a certain number of skills, and a corresponding rune for each, to equip as your skill set. Each skill can have up to 6 different rune versions of it, but you must select which one you want to have equipped at any time. There are no longer different levels of rune abilities, it is simply a matter of having it unlocked or not.
The rune system allows active skills to change in behavior. In some cases, the rune entirely replaces the original skill. For example, the Demon Hunter's Companion skill provides a different companion with each rune. Note however that the passive results of a long running Companion cast are always better with a rune than with no rune. The no rune version does have a distinctive active effect when first cast that may be better for a particular situation than a runed version.
In other cases, the rune adds to the original effect. For example, the Demon Hunter's Hungering Arrow skill has a Puncturing Arrow rune which purely increases the Pierce of the arrow. Puncturing Arrow is strictly better than Hungering Arrow with no rune at all.
You can only select one rune per skill. Runes may decrease some aspects to increase others. For example, the Demon Hunter's Chakram skill has a Twin Chakrams rune which increases the number of attacks while decreasing the damage per attack. It also demonstrates that runes can change the type of damage. In the case of Twin Chakrams, the damage is changed from Physical to Fire.
Changing runes disengages existing casts. For example, the Demon Hunter's Sentry skill runs for a long time (long enough to cast it multiple times). Changing the rune will cause existing sentries to disappear.
You cannot change runes while a skill is on cooldown.
I think of runes as just part of the skill system rather than a system of their own. They increase skill variety without having to select from entirely novel skills.