In Diablo III, Runes are still small objects that must be placed in sockets to take effect, but in Diablo 3 the sockets are in skills, not items. (Items may also have sockets, but runes won't be used in them if they do.) Active skills have sockets in the skill tree interface, and when a valid rune is placed into the socket, the skill gains an appropriate bonus, depending on the type and quality of rune used. Bonuses vary greatly, depending on the skill and the type of rune used, and include such things as extra hits, extra damage, increased spell duration, and many more.
Skill Runes in Diablo III improve a character's active skills in various creative ways. Runes don't just add damage; they modify the skill in some way, making it more powerful and fun to use, as well as allowing for character customization. Runes can be used very strategically; different runes in the same spell will do things. For example: the Wizard skill, Blizzard, would gain different benefits from different runes. A few hypothetical Blizzard bonuses: more falling projectiles, longer duration, more damage per projectile, longer chill time, better to/hit, better chance of critical hit, and so forth. Also, each of those types of bonuses would be increased in power by the quality of the rune used, allowing for a huge variety of potential bonuses and play styles.
As of the BlizzCon demo (October 2008), only some Active Skills had sockets for runes; none of the Passive skills did. Furthermore, no skill had more than one socket. It is not known if any of this will remain that way in the final game.
Runes can be removed from sockets at any time and without any penalty. Theoretically, players could juggle their runes around constantly, sticking in a multi-strike rune to deal with big mobs, changing to a power rune to up the single target damage before a boss battle, and so forth. It's likely that Blizzard will take steps to limit this "rune juggling", perhaps forcing players to return to town before they can change the rune socketed in a spell, putting a cool down time on resocketing the same skill, limiting the number of times a particular rune can be socketed, or even causing runes to lose quality when unsocketed.
Skill Runes will be found in a variety of types. Known runes:
More runes will almost certainly be added during development as the team thinks of useful types to include in the game. The team won't be adding a lot of odd types of runes, though; D3 Lead Designer Jay Wilson has stressed that runes must add a useful function to multiple types of skills. The D3 Team wants all the runes to be useful to all characters, so they won't add runes that just boost one type of skill, or one character's skills. They have to think up rune bonus properties that work across the board, so there are probably not going to be 50 types of runes. (Utility doesn't equal equality, since not every skill can receive exactly the same level of improvement from every rune.)
It's been stated that runes will have various levels of quality, in a fashion somewhat akin to the different quality levels of gems in Diablo II. At BlizzCon 2008, Jay Wilson said there were 5 or 6 quality levels of runes, but that the number could go up or down during development. Higher qualities of runes will (of course) be dropped by higher level monsters, and the higher qualities will be much rarer. The function of each type of rune will not change at higher quality; instead the bonuses will be increased.
The D3 team hasn't yet revealed much information about how item crafting will work in D3, other than to say that they won't have any Horadric Cube-style converting items. We therefore know nothing about the potential for runes to be upgraded by type, though it's certainly possible that something along those lines could be implemented.
At the BlizzCon 2008 demo, runes were stored in an inventory grid on the skill tree menu; not in the normal inventory. There were 10 rune slots below the skill trees. It's not known if they will still be stored there in the final game, if there will be ways to increase the number of slots, or if they can be stored in the inventory instead.