From Civilization V Wiki
There are multiple paths to victory in Civilization V. You can win through scientific dominance, becoming the first civ to create and launch a space ship to Alpha Centauri. You can overwhelm the other civilizations through cultural superiority or political cunning. Or you can employ the ever-popular "crush all of your enemies beneath the wheels of your chariot" tactic and win a mighty domination victory. Whichever civilization achieves a set of victory conditions first wins.
It's important to keep an eye on your opponents' progress toward victory as you advance your own civilization. There's nothing so annoying as to be on the verge of capturing your last surviving enemy's final city, only to watch helplessly as he or she completes her spaceship and wins an upset scientific victory.
How to Win
There are five paths to victory available to you in Civilization V. See the appropriate page for tips and help on each kind of victory condition: