Citizens of the European Union are not required to get a visa for Spain since they already have the right to residency. In order to officially remain resident in Spain you are meant to apply for a NIE (Número de Identidad de Extranjero) and a residency card (Tarjeta de Residente Comunitario). In practice this is not an urgent thing to get done, but you will need it eventually for some things.

Non-EU citizens visiting Spain need a visa (visado) in order to enter and visit Spain, unless there exists a special agreement between Spain and your home country; these countries are those of North & South America in addition to:

Andorra, Australia, Brunei, Costa Rica, Croatia, Cyprus, the Czech Republic, Estonia, Gibraltar, Grenada, Hungary, Iceland, Israel, Japan, South Korea, Latvia, Lithuania, Malaysia, Malta, Monaco, New Zealand, Norway, Poland, San Marino, Singapore, Slovakia and Switzerland.

If you are one of these nationalities, you can enter Spain without a visa and stay for up to 90 days in any 6-month period.

Non-EU citizens coming to work, study or live in Spain are required to obtain a visa - note that this also includes non-EU spouses and dependents. Even if you are not required to have get a visa, in order to officially stay resident in Spain everyone needs to apply for a NIE (Número de Identidad de Extranjero) and a residency card (Tarjeta de Residente Comunitario).