The Spanish Tax year runs from 1st January to 31st December and Spanish residents have to pay income tax (Impuestos sobre personas fisicas) on their worldwide earnings when they complete a declaration (Declaración de la Renta) during May or June the following year. A Spanish resident is defined as someone who spends more that 182 days in the year in Spain not whether the citizen has a residence card or not.
Salaried workers earning less that 22.000 Euro a year probably will not have to complete an income tax declaration as long as they have a single employer. The Tax Agency will send you a form 105 to authorise that you agree with the calculations and return. Additionally you do not need to fill in a declaration at all if your income is less than 8.000 Euro a year. Everyone else files a declaration between May 1st and June 30th and either pays the tax due or claims a "devolucion" which they will receive in a few months by bank transfer.
The tax agency have published a computer program to complete the Annual declaration called PADRE. Many banks and Gestores now use this program. Your professional advisor can help you with the filing and will charge 70 euros upwards depending on the complexity of your case. Married couples should check whether it is better to file joint or separate declarations.
Spain has double taxation agreements with many countries including all European and Scandinavian so these citizens will not be taxed twice. Americans will still have to file a tax declaration because the USA bases tax on nationality not residence.