The Generic Universal RolePlaying System, usually shortened to GURPS, is a roleplaying game made by Steve Jackson Games. It has loads of numbers and expansion books. GURPS is the quintessential universal system; it is so flexible that you can bend it in half, fold it through itself, and then tie it in a 4-dimensional knot. It is the paragon of the simulationist category of games. All simulationist games since GURPS secretly aspire to kill GURPS and wear its skin while drinking the blood of its delicious heart. Its detail is surpassed by I.C.E.'s Rolemaster series, but GURPS doesn't suffer from the Table Within a Table Within a Table Within a FUCK YOU problem inherent in the game of Rolemaster.
The basic premise behind the system is that you create your character and customize all of his abilities using points that you get both at creation and as your character progresses. There are assloads of physical, mental, and social defects that can be used to get more character points to channel right back into your advantages. It uses the d6 exclusively, generally pressuring you to roll under your relevant ability score to succeed in a given task. This can be augmented in difficulty by increasing or decreasing the target numbers. A roll of a 3 or 4 is a critical success, while an 18 is always a critical failure. SJG put out a free .pdf synopsis of the rules called "GURPS Lite", which is enough to play a barebones version of the game.
GURPS' simulationist bent results in it being regarded as excessively complicated by many gamers, earning it the humorous moniker "Generally Unplayable RolePlaying System". Using its third edition vehicle creation system will make you want to eat your own face, and possibly devote the whole process to one or more elder gods. (Fourth Edition vehicles are just regular characters with extra stats.) Game Masters of the franchise experienced a downgrade in complexity following the release of Fourth Edition, but its inherent complexity still puts it behind Dungeons and Dragons in terms of ease of play.
Much of the complexity in GURPS is front-loaded: character creation tends to take longer because of the staggering amount of detailed options. Forget just rolling a character like you would in D&D, you need to set aside a session to build one. That's the price you pay to evade Linear Build Quadratic EXP.
As the campaign progresses, however, the consistency and relative elegance of GURPS makes Dungeons and Dragons seem like a pile of kludge. The sandbox nature of character creation is appealing to players who have a concept of a character in their heads and want to reproduce it as faithfully as possible.
Generally, anything can be done with just the base starter set of GURPS and a good head on your shoulders, but if you're too busy to do that the guys at SJ Games have it all figured out for you ahead of time. All of the books for GURPS simply tell you how to do things within the core system without the need to spend an additional day to set up an innovative campaign. For example, there is a book on Vampires. It explains how to create vampire characters and NPCs easily without dealing with the issues of balance and customization that bogs down new players. There is even a Dungeon Fantasy book you can get off E23, the Warehouse 23 PDF site, that allows you to play Dungeons and Dragons without the need to buy polyhedrals. In fact, you can play this game by simply using the dice you swiped from Monopoly and Yahtzee. One irritating fact you'll discover as you collect GURPS books is that although the system uses real-word measurements, the different authors have never agreed on whether they should use Imperial or metric units. Keep a conversion chart handy. Fortunately the main book includes one.
In the game's rather long lifespan, a truly impressive amount of settings and sourcebooks-usable-with-many-settings were made, including but not limited to Alpha Centauri, settings where you play Men In Black, Fantasy (including Discworld), multiple SF settings (notable licensed one: GURPS Vorkosigan Saga), several books on conspiracies, historicals covering WW2, Ancient Rome, multiple other ancient cultures, joke settings (GURPS: IOU illustrated by Phil Folgio!), adaptations of a couple other games, (Notably the Classic World of Darkness (Vampire, Mage, and Werewolf), Traveller and Castle Falkenstein) and a couple books that are just collections of multiple settings. For starters. If you've got a group who feels like playing something different every month, GURPS has got you covered for the next decade or so. New stuff occasionally still comes out, but it's slowed to a trickle compared to the old days.