Oblivion:Personality

Personality is the governing attribute for Mercantile, Speechcraft, and Illusion. It affects the Disposition of NPCs and Creatures, increasing Disposition by 1 point for every 4 point increase in Personality.   Therefore, Personality allows your character: 
To more easily obtain meaningful information from NPCs.
To obtain better prices from merchants through haggling.
To avoid combat more often. Fewer aggressive beings will attack you and more will accept yields.

In-game Description: Affects how much people like you, which leads to better information-gathering. Governs the skills of Speechcraft, Mercantile, and Illusion.

Character Creation

The Races page provides a table summarizing the initial personality for each race and gender.

No birthsigns provide personality bonuses.

The following standard classes have personality as a class attribute (initial +5 bonus):
Agent, Bard, Healer, Knight, Pilgrim, Rogue

Or any custom class can select personality as one of the custom class' two attributes (although this is not recommended, see Notes).

Notes
The primary advantage of high Personality is that it increases dispositions.  However, several other methods, such as Charm spells, are also available for increasing dispositions, as detailed on the Disposition article; these other methods can often be more effective than moderate increases in Personality.  Personality becomes more effective if you can achieve very high values.
Increasing Personality past 100 (for example, with Fortify Personality) continues to improve your disposition scores for NPCs and Creatures.
With a sufficiently high Personality, normally hostile beings (e.g. bandits, rats, dremora, etc.) will no longer attack you on sight, because their disposition is so high it counters their high aggressions.  Your personality needs to be roughly 180 (depending upon Fame and other factors) for Animals to become non-aggressive; higher Personality is necessary for other enemies.  Enemies may still attack your Followers.
While Personality is the governing attribute for the Illusion skill it has no impact on actual Illusion spells cast (see Magic Overview).
If you are planning on attaining a high Personality, it is recommended that you do the Namira's Shrine Quest as early as possible, as the quest is harder to initiate the higher your Personality.

See Also
Increasing Attributes: Tips on how to increase personality.
Magic Items: There are multiple generic items with that fortify personality.
Unique Items: A list of unique items, several of which fortify personality. These do not include Leveled Items such as those received from quests, or Artifacts.
::*Imperial Breeches are a pair of unique pants that can be obtained from Thoronir at the Copious Coinpurse. Among other things the pants fortify your personality by 5 points.
::*Robe of Creativity is a unique robe that can be obtained from Nilawen at The Fair Deal. Among other enchantments, the robe fortifies your personality by 5 points. Note that, although identified as a robe, this item only covers your upper body, and therefore can be worn with pants or greaves.
::*Vest of the Bard is a unique shirt that is obtainable from Gunder at the Colovian Traders. Among other things the shirt fortifies your personality by 5 points.