Working in Cyprus
A work permit is required. This is a formality and can easily be obtained by EU citizens by aquiring approval from the Ministry of Labour.
Non-EU citizens need to contact the Migration Officer in Nicosia to apply for a work permit.
For both non and EU citizens, your employer can make the application.
Looking for work in Cyprus
The best place to start are the local newspapers. There are also various job agencies on the island which can help.