As an EU citizen, you will have the same right to work while studying as nationals of that country. You will not need a work permit, even to work full-time.
Exception - Croatian nationals still face temporary restrictions on working in the EU.
More on temporary working restrictions.
Several countries allow you to work an unlimited number of hours per week. Others apply limits per term, or per year.
More on working in another EU country.
If you live in another EU country for more than 6 months a year there (183 days) you will be considered tax-resident there. Consequently, if you work while studying, you will have to pay taxes and social security contributions on income earned in the country where you study.
The country where you study could also tax income earned in a different country, for instance from any summer jobs you may get in your country of origin. You may have to report such income and, in some cases, pay tax on it in the country where you study.
Be aware that many - but not all - EU countries have agreements to avoid double taxation. These agreements may in some cases determine which country you should pay tax to.
More on taxes.
Joost is a Dutch student who moved to the UK to study. He works part-time at a restaurant on a contract under English law. His employer deducts taxes and social security from his monthly pay checks and pays contributions for him to the UK authorities.
During his summer holidays, Joost works full-time in his native town in the Netherlands. His employer there pays contributions for him to the Dutch tax authorities. Joost, who is a UK resident for tax purposes, can either request a refund from the Dutch authorities and then pay the applicable income taxes in the UK, or declare the income earned in the Netherlands and the taxes already paid there to have a tax discount from UK authorities instead.