Great People are powerful civilian units that cannot be conventionally produced. Instead, they are created after accumulating enough Great People Points, which can be done by constructing certain buildings or wonders and assigning specialists. Each type of Great Person has an individual progress bar, which once filled will generate that type of Great Person. However, each Great Person generated will increase the amount of points required to generate the next Great Person of any type. Great Generals, Great Admirals and Great Prophets are an exception to this rule. Another thing to note is that all Great People, including Great Generals and Great Admirals, are not considered combat units, and will be destroyed if an enemy unit moves to the tile they're on. All Great People have two movement points, with the Khan being the sole exception.
All Great People have the ability to instantly start a Golden Age, which will consume the Great Person. However, this ability is limited to the Great Artist in the Gods and Kings expansion. Also in Gods and Kings, each type of Great People can be purchased with Faith once the Industrial Era starts and the relevant Social Policy branches are filled out. For example, finishing Commerce will let you purchase Great Merchants while finishing the Freedom branch lets you purchase Great Artists.
Great Artists have abilities related to Culture. They can Culture Bomb tiles to add them into your territory, though doing so near other civs' borders will likely anger them. They can also create the Landmark improvement, which grants a high Culture yield from that tile. In Gods and Kings, they become the only type of Great Person to be able to generate Golden Ages, while their Culture Bomb ability is removed. Using the Great Artist in any of he stated ways will immediately consume them.
Great Engineers have abilities that help increase Production. They can use Hurry Production to quickly generate huge amounts of Production, which is very helpful when rushing an important Wonder. They can also create the Manufactory tile improvement, which grants a high Production yield when worked. A Great Engineer is consumed when used in either of these ways.
Great Merchants have abilities that boost your Gold supply. They can embark on a Trade Mission to a City State to generate a ton of Gold quickly. They can also create the Customs House tile improvement, which grants a high Gold yield when worked. A Great Merchant is consumed when used in either of these ways.
Great Scientists have abilities geared towards Science. They can grant you a free Technology, which can be vital in the later stages of the game. This ability is replaced with one that grants you as much Science as your empire has generated in the past 8 turns. They can also create the Academy tile improvement, which grants a high Science yield when worked. A Great Scientist is consumed when used in any of these ways.
Great Generals improve your military units. Unlike other Great People, they are generated after enough experience have been obtained from warring with other civs. Their presence will grant a 15% combat bonus to all your own land units within a two tile radius. They can also create the Citadel tile improvement, which will grant a great defensive bonus to friendly units inside and cause any adjacent enemy units to take 3 damage (30 in Gods and Kings). Citadels have a similar function to the removed Culture Bomb ability in Gods and Kings, in that they will convert tiles around them into your territory. A Great General is only expended when used to build a Citadel of when starting a Golden Age.
The Mongolian civilization has a unique replacement for the Great General called the Khan. In addition to a normal Great General's abilities, the Khan has a higher movement speed, as much as a Keshik. They also heal adjacent units for 2 hp (15 in Gods and Kings) each turn.
Great Admirals (G&K only)
Great Admirals improve your naval units similar to how Great Generals improve land units. They are generated after enough experience have been earned by naval units from warring with other civs. Like the Great General, Great Admirals give a 15% combat bonus to all of your naval units within a two tile radius. They can also use Repair Fleet to completely heal the naval unit that shares its tile and all adjacent naval units. A Great Admiral will be consumed after using Repair Fleet.
Great Prophets (G&K only)
Great Prophets are vital for creating a religion and ensuring it spreads well. They are generated randomly after enough Faith is accumulated, but must be purchased with Faith after the Industrial Era. You can also capture enemy Great Prophets, the only type of Great Person that isn't destroyed when you move on to their tile. They can found a religion, provided that it's still possible to do so, enhance a religion you've founded or spread your religion similar to a Missionary. They can also create the Holy Site tile improvement, which grants a high Faith yield when worked. A Great Prophet will be consumed when used in any of these ways.