Science

Science is a Fallout, Fallout 2, Fallout 3, Fallout Tactics, Fallout: New Vegas, Van Buren, and J.E. Sawyer's Fallout Role-Playing Game skill.

Fallout, Fallout 2, and Fallout Tactics

 * $$\text{Initial level}\%=25\%+(2\%\times\text{Intelligence})$$

Example: A starting Intelligence of 5.
 * $$25\%+(2\%\times5)=35\%$$

The Fallout Tactics 1.27 patch notes readme file says:

suggesting that Science somehow affects critical chance or effectiveness against robots (at least in Tactics).

Fallout 3

 * $$\begin{matrix}\text{Initial}\\\text{level}\end{matrix}=2+(2\times\text{Intelligence})+\left\lceil\frac{\text{Luck}}{2}\right\rceil$$

Example: A starting Intelligence of 5 and Luck of 5.
 * $$2+(2\times5)+\left\lceil\frac{5}{2}\right\rceil=15$$

Science primarily allows you to hack terminals using a mini-game. Certain terminals have different difficulty levels, and different Science skill levels are required to hack them. The difficulty ratings and skill levels are: very easy (15), easy (25), average (50), hard (75), and very hard (100). Your Science skill relative to the difficulty rating affects how hard the minigame is (at Science 100, average terminals are a breeze, presenting only about 5 possible passwords). Harder terminals have more words, longer words, and fewer brackets.

Ways to increase Science

 * Permanent
 * Levelling up (up to +15)
 * Bobblehead - Science (+10)
 * Big Book of Science (+1 or +2 with Comprehension)
 * Daddy's Boy/Girl (+5 for each rank, max 3 ranks)
 * Tag! (+15)
 * Survival Expert (+2, 4 or 6)
 * Covert Ops (+3) (Operation: Anchorage)
 * Temporary
 * Vault lab uniform (+5)
 * Enclave scientist outfit (+5)
 * Lesko's lab coat (+10)
 * Scientist outfit (+5)
 * The Surgeon's lab coat (+5)
 * Mentats (+2-10 depending on Intelligence, a perfect Intelligence of 10 will see no benefits)
 * Lab coat (+5) (Broken Steel)

Interactions that require Science skill

 * 50 Science to reactivate the robots during Big Trouble in Big Town.
 * 60 Science to unlock certain dialog options with Doctor Lesko during the quest Those!. This allows you to learn he had been using a modified Forced Evolutionary Virus to induce pyrosis reactions in giant ants, creating fire ants.
 * 75 Science to modify healing archways to fully heal the player, though this reduces their number of uses to three. (Mothership Zeta)
 * 75 Science to hack the Mobile Crawlers' mainframe. (Broken Steel)
 * 80 Science to deactivate the repulsion fields aboard the Mobile Base Crawler. (Broken Steel)
 * With a high Science skill you can bypass the speech check required to convince John Henry Eden to self destruct during The American Dream.
 * Although a high Science skill increases your chances of disarming a slave collar, the Explosives skill is also taken into account. Even when both skills are high, the slave could still blow up, so saving before attempting to disarm a collar is advised.

Fallout: New Vegas

 * $$\begin{matrix}\text{Initial}\\\text{level}\end{matrix}=2+(2\times\text{Intelligence})+\left\lceil\frac{\text{Luck}}{2}\right\rceil$$

Example: A starting Intelligence of 5 and Luck of 5.
 * $$2+(2\times5)+\left\lceil\frac{5}{2}\right\rceil=15$$

Ways to increase Science

 * Permanent
 * Levelling up (up to +15)
 * Big Book of Science (+3 or +4 with the Comprehension perk))
 * Temporary
 * Programmer's Digest (+10 or +20 with the Comprehension perk))
 * Followers lab coat (+10)
 * Scientist outfit (+5)
 * Night Person (+4)
 * Mentats (+2-4 depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * Party Time Mentats (+2-4 depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * All-purpose science suit (+5)

Interactions that require Science skill

 * 25 Science to use the chemistry set in Mitchell's House.
 * 25 Science to discover that Victor has been overridden when you asked him to help fighting the Powder Gangers in Goodsprings.
 * 30 Science to reprogram Primm Slim to become the Sheriff of Primm.
 * 45 Science to obtain the Rebreather without finding a pressure cooker for Jack in Volare!.
 * 50 Science to hack Daniel Contreras's terminal and download the weapons manifest to get the weapon This Machine.
 * 50 Science to figure out who hacked Primm Slim in Primm.
 * 55 Science to upgrade the rocket trajectory at the end of the quest Come Fly With Me.
 * 55 Science to run the bypass on ED-E, thus lowering the Repair requirement to 35 (instead of 65). However on the Xbox 360 (no patches) this is reversed, 55 Repair lowered to 35, requiring 65 Science.
 * 60 Science to repair Rhonda in Black Mountain.
 * 70 Science to convince Keely that the player must return the data of Vault 22 to Thomas Hildern in the quest There Stands the Grass.
 * 80 Science to bypass the fee to enter The Strip.
 * 80 Science to convince Scribe Ibsen to delete the Brotherhood of Steel's files on the computers at Hidden Valley.
 * 90 Science to convince Henry to test the Stealth Boys on nightstalker brains in Guess Who I Saw Today.

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