It's empadronarse (one of those lovely reflexive verbs!), which means to register -- but it seems to be primarily used for registering with the city. To do this, you need to go to the local office of your town hall -- el ayuntamiento. If you don't know where your local office is, go to the main town hall and someone there should be able to tell you. Most ayuntamientos also have a web site where you may be able to search for your local office. There you will need your passport or DNI and proof that you are living where you live -- either a rental contract, a notarized letter from the owner stating that you live there or piece of official mail, such as a bank statement or an electric bill in your name.
As for social security, you should go to the local branch of your social security office. To find out where yours is, you can go to the English version of the
Seguridad Social web site and click on ADDRESSES AND TELEPHONE NUMBERS or click on ENQUIRIES to send an email.
Buena suerte,
