Not all European nations grant citizenship through grandparent lineage; before beginning this somewhat process, verify that the nation of which you seek citizenship will allow you to claim it through your grandparent.
If the country does in fact accept this type of application, make the connection. Find out all the information concerning your European grandparent and gather all documents surrounding her birth and immigration. Obtain your grandparent's birth records from the records department of the country, city and/or county of her birth if you do not already possess them. If necessary, obtain your grandparents' marriage certificate and immigration records or a copy of your European grandparent's green card or passport.
Gather documents concerning your birth, as well as the birth of your parents and their marriage certificate. Obtain proof that your parent is the child of the EU citizen.
Research the application procedure, and acquire the specific form for citizenship of the nation of your connection. Locate the embassy or consulate general website online to understand procedure and complete the application forms. Some countries, such as Ireland, allow application for citizenship completely through the mail. However, many EU countries will require an in-person interview before granting citizenship.
Send or bring your application and supplemental documents (marriage certificates, birth certificates, etc.) to the embassy or consulate general of your intended new country of citizenship that has jurisdiction over your area of residence. Do not make the appointment until you have all of the necessary forms and documents in hand.
Pay the application fee. All applications for citizenship require a processing fee; the amount varies with each country.
Apply for a passport. Once you have received citizenship of a country, you may apply for a passport following the application procedures of the country.