Members of the SCA participate in events in persona or who they would have liked to have been in the Middle Ages. How developed your persona is up to you. At a minimum, people choose a name, a culture, and a time period. You will meet many people who have only gone this far in developing their persona while others have created an extensive biography of their persona.
Choosing a Name and Creating a Persona by Modar Neznanich is a great resource to help you in developing your persona.
Choosing a Society Name: Hints for Newcomers by Dietmar Reinhart von Straubing and Malachias von Morgenstern is a good resource for how to choose your name.
There have been many members of the SCA who have done extensive research on the names used during the Middle Ages. They have been compiled into the Medieval Names Archive. If you choose a name from one of these articles, then registering your persona's name with the SCA will be easier than documenting the name yourself.
Sir Koppel, a member of the Barony of An Crosaire, also researches and collects names.