You want to stay and work legally in the Netherlands after your study. Non-EU nationals have to meet more legal conditions than EU-nationals. These are briefly listed below. 

EU-nationals 
Residence Permit : EU-nationals don’t need a residence permit for the legal stay in The Netherlands. However, a residence permit can be useful for other purposes than legal staying. For example: to be a member of a housing-corporation or to vote. 

Work permit: Employers don’t need a work permit for EU-nationals. There is one exception: employers of EU-nationals of Bulgaria and Romania need a work permit. Note that it is your employer who needs one, not you. 

Non-EU nationals, residence permit and work permit
There are three possible residence permits to stay in the Netherlands after your study. 

1. Residence permit for ‘seeking work after graduation’.
Condition: Bachelor or Master diploma. Duration: one year.
This is the easiest one to obtain. There are no conditions regarding your income. You just show your diploma to IND.

2. Residence permit according to the ‘admission scheme for highly educated persons’.
Condition: Master diploma of PhD. Your gross annual salary must be at least €26,931 (2012) Duration: one year, cannot be extended.

3. Residence permit for work.
Condition: Master diploma and employment contract. Duration: equal to your employment contract. In this case your employer needs a work permit for you. Maybe he qualifies for a work permit for you as a performing artist or musician. 

Health insurance
As a student without work, you are not obliged to have a Dutch Health Insurance. You can have your own health insurance. However, as soon as you work or as soon as you are not registered as a student any more, you should have the Dutch Zorgverzekering. This is a legal obligation. You risk a fine equal to 130% of the premium of the non-insured months.