EU/EEA members (plus Iceland, Liechtenstein, Norway, and Switzerland nationals) can work and live in Greece without a permit. They are only required to register with the authorities in Greece. Additional information can be found at: http://www1.mfa.gr/en/visas/
Aupairs from outside of the EU must contact their local Greece Embassy to see if they qualify for a work permit: http://www.embassyworld.com/embassy/Greece/Greece.html
Resident permits are very difficult to obtain. They require Government approval, a large deposit, by aupair or employer. Very few are approved. Greece has the lowest approval rate of any resident permit. You need to first obtain a visa to enter Greece then apply for a permit within 30 days of your arrival if you intend to work there longer than 90 days. The process from start to finish, could take up to a year.