There are five main types of victories in the game: Domination, Science, Culture, Religious, and Score Victories. 

Domination Victory - can be achieved by having control of all original capitals in the game and other players who still has their original capital, would need to capture all the other original capitals that you have taken.

For Domination Victory, I don't believe that second sentence is true as the other player with their original Capital would need to take all of the other original Capitals that you conquered.

Science Victory - can be achieved by three steps: Launching a satellite, landing a human on the Moon, and establishing a colony on Mars. Launching a satellite requires researching the Rocketry Tech, building a Spaceport, and completing the Earth Satellite project. Landing a human on the Moon requires researching the Satellites technology and completing the Moon Landing project. Establishing a colony on Mars requires researching the Robotics, Nuclear Fission, and Nanotechnology techs and completing each respective module project. Once all three module projects are built, you will have won the game. 

Culture Victory - can be achieved by having a total number of tourists that exceeds the domestic tourists of every other civilization in the game. You can achieve these tourists by having multiple trade routes, shared/conflicting late game Governments, open borders, Great Works, Relics, Holy Sites, National Parks, Artifacts, and wonders.

Religious Victory - can be achieved by converting every single other civilization to your Religion, which can simply be achieved by having the majority of their cities following your Religion. This can usually be achieved by having a high number of Apostles and Missionaries.

Score Victory - is a special victory that can be obtained after the year 2050 or whichever turn the player sets. It is determined by a civilization's score, a score that accumulates over the course of the game from your number of civis, number of population, number of cities, number of technologies, number of future technologies, number of wonders, military accomplishments, number of Great People, and number of Beliefs.