Oblivion:Willpower

Willpower is the governing attribute for Alteration, Destruction, and Restoration.  It affects:
Regeneration rate of Magicka (see below).
Fatigue = Willpower + Agility + Endurance + Strength

In-game Description: Affects how quickly you regenerate Magicka, and how much Fatigue you have. Governs the skills of Destruction, Alteration, and Restoration.

Character Creation

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

An initial +10 willpower bonus is possible by selecting Lady as birthsign.

The following standard classes have willpower as a class attribute (initial +5 bonus):
Crusader, Healer, Mage, Monk, Nightblade, Spellsword

Or any custom class can select willpower as one of the custom class' two attributes.

Magicka Regeneration
The magicka regeneration rate appears to be linear in Willpower, and unaffected by Luck. The formula is:
 MagickaRegenRate = 0.01 * (fMagickaReturnBase + fMagickaReturnMult * Willpower) * TotalMagicka.
Where the construction set specifies that  =0.75 and  =0.02.

For example, with 100 base Magicka and 100 Willpower, your Magicka regeneration rate will be 0.01 * ((.02 * 100 Will) + .75) * 100 Magicka = 2.75 Magicka points per second.  The regeneration rate will continue to increase if your Willpower is increased past 100.

Therefore, your regeneration rate is affected by two factors: the size of your Magicka pool, and your Willpower.

Willpower basically determines the seconds spent to regenerate 100% of Magicka. The formula is:
 Seconds = 100 / (fMagickaReturnBase + fMagickaReturnMult * Willpower).
For example, with 100 Willpower your Magicka will completely replenish in 36 seconds, no matter if the pool is 100 or 1000, it takes always 36 seconds. 
With 0 Willpower, the time is 133 seconds, so 0 Willpower won't stunt your Magicka.

The regeneration rate can also be expressed as a percentage of the total Magicka pool.
 Regen % = fMagickaReturnBase + (fMagickaReturnMult * Willpower)
With the values specified by the construction set, this solves as 0.75% total Magicka regeneration per second, increased by 0.02% per point of Willpower. Therefore, 100 Willpower results in 2.75% total Magicka regen per second, corresponding to the previous examples.

A character with a weak Magicka pool and a high Willpower may not regenerate raw Magicka as quickly as a character with a strong Magicka pool and a moderate Willpower.  This is what makes racial and birthsign Magicka bonuses more powerful than they may appear at first glance.

The Stunted Magicka effect of both the Atronach birthsign and certain diseases (such as Astral Vapors) will prevent Magicka from regenerating from Willpower.

Intelligence vs Willpower
Improvements to either Intelligence and Willpower will increase your character's Magicka regeneration rate.  If you need to choose between an increase to Intelligence and an increase to Willpower on leveling, one factor may be which increase will have a greater effect on your Magicka regeneration rate.

At low levels, boosts to Intelligence will be more effective than boosts to Willpower of the same magnitude.
At high levels, boosts to Willpower become more effective.  The exact crossover point occurs when your total Magicka (including both Intelligence and racial/birthsign bonuses) becomes 75 points larger than two times your Willpower.
For example, if your maximum Magicka is currently 150, and your Willpower is 50, then you should keep increasing Intelligence rather than Willpower for another 13 points (until your maximum Magicka exceeds 175).
This comparison is based only upon the regeneration rate.  However, increasing Intelligence always increases your total available Magicka, too.    Some players may prefer to continue increasing Intelligence even at high levels.
If comparing the relative effect of attribute increases of different magnitudes (for example, +2 Willpower vs +3 Intelligence), you will probably need to use the above formula to calculate the exact effects. 
Increasing both Willpower and Intelligence simultaneously will, of course, have the greatest overall effect.

If you want to have your character regenerate magicka at the maximum rate possible in the game, it is best to use enchantments that increase your total magicka rather than increase your willpower. Using a "normal" character (one with no applicable sign or racial bonuses) with 100 intelligence and 100 willpower, you will have 200 magicka and a regeneration rate of about 5.5 points/second. If you use the best possible enchantments to raise willpower (9 pieces of equipment that have been enchanted with a +12 willpower Sigil Stone) you will have 208 willpower, still have 200 magicka, and regenerate 9.8 points/second. If instead, you use 9 pieces of equipment enchanted with a +50 magicka Sigil Stone, you will have 650 magicka, and regenerate it at 17.9 points/second. An Altmer character with The Apprentice sign would yield regeneration rates of 23.4 points/second if maximizing his magicka and 19.6 points/second if maximizing his willpower.

Notes
Although there is a loading screen hint that claims "A high Willpower allows you to defend against magical attacks," no evidence has been found from gameplay to support this claim.  It may be a residual message from earlier Elder Scrolls games, in which Willpower provided resistance to effects such as paralyze and silence (see Morrowind for example). 
Players who take the Atronach birthsign have no need to increase willpower other than for the purpose of increasing their Fatigue score.
If you have a total Magicka pool of 275 or greater and have to choose between putting points into Intelligence or Willpower you will gain a faster Magicka Regen Rate by putting them into Willpower until it is maxed.

See Also
Increasing Attributes: Tips on how to increase willpower