There are multiple ways to achieve victory and, provided that they aren't disabled in the advanced options, being the first to achieve any will let you win the game.
A Cultural Victory is achieved when you build the Utopia Project, which requires you to have completed five Social Policy branches. Social Policies cost Culture to adopt, and becomes more expensive with each founded or annexed city. This makes Cultural Victories more desirable for tall empires, which consists of only a few, very well-developed cities.
A Domination Victory is the most straightforward, in that you need to be the last civilization to keep your original capital. You don't have to be the one to capture all the other capitals, in fact you can potentially win if one civ that has lost their capital captures the capital of the original capital of another civ. Dominaion victories require you to build a strong military force, and certain civs like Japan, The Aztecs and Mongolia are geared for it.
To win through diplomacy, you must get the majority vote from the United Nations. The civilization that constructs the United Nations will get an extra vote, giving them an advantage. To garner votes, you must become allies with city states, or liberate the city of an extinct civilization or a city-state. It is easier to do the former than the latter, so aiming for a Diplomatic Victory will usually result in an increased focus on your economy.
A Scientific Victory requires you to build a space ship, thus it is commonly referred to as the space ship victory. First, the Apollo Project must be constructed to start the space race. Then, you must research the technologies required to build each part of the space ship. Once you have constructed all the parts, you must transport them to the capital, where they will automatically be assembled and launch, winning the game.
A Score Victory is pretty much never aimed for until very late in the game, when it seems impossible to win before the maximum number of turns run out. Once a certain period has reached, by default the year 2050, the player with the highest score will win. The score is determined by a lot of factors, from amount and size of cities to military power to level of technology to your financial strength.