A European passport is a great asset that will allow you to travel and work freely in an European country. If you are married to a EU citizen, you may very easily obtain one. Although there is no fixed common European passport, if your spouse is a citizen of any of the European Union member countries, you may obtain a passport from that country and, with it, the rights and benefits of being a European citizen. But the individual process to obtain a European passport through your spouse will vary based on her country's laws. This is meant as a general guide, but, if for example your spouse is Italian, you should check current Italian laws on obtaining Italian citizenship (and passports) through marriage.
Apply for citizenship of your spouse´s country. The procedures for doing so will vary greatly depending on the country your spouse is from. For example, while Spain requires a minimum of two year´s residency in the country as a couple, Italy requires the same, but adds an exception for couples that have been married more than three years and living abroad. Germany requires a marriage of at least two years, with the foreign spouse having lived in Germany for at least three years. You can obtain more information at the closest embassy or consulate to your place of residence. But do expect it to be a rather lengthy process.
Formalize citizenship requirements. Many European countries will request you formalize your citizenship by becoming an official resident of the country, applying for social security and therefore paying taxes in that country, or other such procedures. For example, Germany will require you to "give up" your original citizenship in order to obtain the German one, and swear an oath to the German constitution as well as prove your knowledge of the German language, after demonstrating that you were residing in Germany as an official resident, and proving that you are financially self-sufficient. Spain will require that you present a certificate of good conduct issued by the police, and letters from two Spanish citizens attesting to your good citizenship and integration into Spanish society. You will be informed of your spouse´s country regulations concerning formalizing your citizenship, when you apply for it.
Once you obtained the citizenship through your spouse, you can apply for a passport for that country. Again, you will have to gather the necessary paperwork and follow the procedures that the local embassy or consulate will give you, since this step also varies on a country by country basis. This process is usually faster than the citizenship process. For example, after you obtain your Italian citizenship, if you are abroad you will go to your closest Italian consulate and fill out an application for it after presenting the documents stating that you are a citizen. If you are in Italy, you will take the same documents, but to the local police headquarters in order to obtain your passport. In Spain, you will have to make an appointment at the closest police headquarters that offers passport processing, and if you just bring the documents granting you citizenship and a Spanish ID card on the date and time of your appointment, you will walk out with a Spanish passport in less than 30 minutes. If you are abroad, but obtained a Spanish citizenship, you may obtain your passport through the consulate that did your Spanish citizenship paperwork.