The Greek Gods are the caretakers and rulers of western civilization. Since the beginning of time, the gods have ruled and governed the universe. Their power is drawn directly from western civilization.
They were not the first guardians of the universe, though. Prior to them, there were five ages of history of which the world was governed. However, the world as is is now functions because the gods and their whims influence every part of society and nature.
The first gods are the children of Kronos and Rhea--the king and queen of the Titans. These gods--Hestia, Hades, Demeter, Hera, and Zeus--were prophecized to overthrow their dictator of a father and claim the world for themselves. As such, their father devoured all but Zeus, who escaped thanks to his mother's assistance.
When Zeus returned to meet his father, he freed his siblings and they went to war with the titans. This was the beginning of the First Titanomachy, which lasted eleven years. It ended with the gods winning and claiming the world for themselves.
Gods are different from all other forms of being in that they can be created and disempowered through the joint power of the Olympian Council. Though it is unclear how the Elder Gods came into existence--they are the only offspring of two titans to not be titans--all those born to a god and another immortal being has proven to be a god. Whether they bore the children of other gods, titans, or nature spirits, the result has proven to be gods.
Gods have the ability to bestow immortality and power onto anyone whom they deem worthy. This is most commonly used as a reward for great services to they Olympians or to marry a lover. As such, many of the immortals who were not always gods are either heroes of old or the consorts of their Olympian patron.