In Fallout, Luck primarily affects your chances with the virtual dice of the game: a high Luck means you're more likely to succeed a roll, while the opposite is also true. Luck also affects the chance to score a critical hit on an enemy, with the standard being that your Critical Chance is equal to your Luck (keep in mind, standard means without perks, traits, or aimed shots). It also determines the initial level of your Gambling skill.
The Hubologists can be visited to gain a Luck bonus. Only one bonus can be gained, though they have two bases, one in the NCR and one in San Francisco. The NCR bonus can either boost Luck by 2 or reduce it by one, and the San Francisco base can either boost Luck and Intelligence by 1, or reduce Luck by 1.
Taking the Gain Luck perk for an increase of 1 Luck.
If you have the Hubologist zeta scans, it will stop the decrease in luck from the Pariah Dog joining your party. Subsequently killing the pariah dog will then restore your Luck as if it had been decreased, resulting in a gain in Luck.
Modifies: All Skills and Critical Chance.
Unlike other S.P.E.C.I.A.L. scores, Luck has no specific skills associated with it. Instead, every two points of Luck adds one point to all skills. Having a high Luck will also improve your Critical Chance with all weapons.
Luck functions in the same manner as Fallout 3. It raises the value of every skill (see chart at bottom of page) and affects Critical Chance. It also improves the odds of winning at any casino game; the higher the Luck, the more likely favorable outcomes on bets will be. If you win a casino bet through the influence of Luck, you receive a "You feel lucky" message. A luck of 6 will begin to influence Roulette and Slot machines, though rarely. A luck of 7 is required to win with any regularity or to begin to influence Blackjack outcomes.
If the player's Luck stat is the most extreme (highest or lowest), Doc Mitchell will either say "With luck like yours I'm surprised them bullets didn't turn right around and climb back into the gun." (for highest) or "Now that don't make a lick of sense. Seems to me you're the luckiest son-of-a-gun in New Vegas." (For lowest)
When Doc Mitchell refers to the bullets "climbing back into the gun" this is a notion to Fallout 1 and 2, referring to Critical Failures.
When playing at casinos, a Luck of at least 7 is required to get favorable odds. Blackjack is the most affected by Luck; at 10, it will produce fairly constant patterns in what cards are dealt to you based on your current total. Slots will turn out losses on most spins, but at higher Luck will produce an occasional jackpot. Roulette, like Slots, will produce a loss on most spins and requires prudent betting to gain much benefit from Luck.