For EU citizens
First make sure you are a citizen of the EU, then you can start looking for a job. There are different organizations that were created to help you, for example EURES (European Employment Service) aims to facilitate the free movement of workers within the countries of the European Economic Area, but there are more organizations aiming to help you.
If you are part of the original EU states you should be able to work in an EU state without restrictions. If you are part of a newer EU state, or you wish to work in a newer EU state, check the work restrictions with that country.
Request a passport if you do not already have one. No matter your nationality, you should apply for a passport, in certain cases you will be required to have this document.
Check out the European Commission website for jobs, if you are a citizen of the EU. The European Job Mobility Portal, lists job openings, tips for living and working in the country.
For non-EU citizens
You should look for areas that you may potentially be hired based on the languages you speak and the job you perform. You should definitely apply for jobs or visas before going abroad.
If you are still studying here is a useful tip if you want to work abroad later: get a degree in a field that is currently in demand in Europe. You might be hired easier or you may not be required to have a European work permit at all.
Non-EU citizens may need to obtain a work permit and a resident visa before they can work in Europe legally.