As an independent, you belong to no faction in particular. Your character has a lot of personal freedom to do what he or she wants with their life. You will be given enough credits to get you started, and a choice of worlds to begin on. Where you go and what you do from there is up to you.
For most new independents, the first priority is finding a ship. Ships are expensive, and you probably won't be able to afford one with the money you start off with. You have a number of options, however. You can hire on as crew with the captain of an existing vessel to earn some money. You can take out a loan from a bank or wealthy player, or perhaps find a faction willing to sponsor you. Or, you could pool your cash with other independent characters and form a partnership.
Whatever you decide, the key to being an independent is making things happen. Adventure and excitement aren't going to come to you, you'll have to go out and find them. An independent who waits around, hoping to be in the right place at the right time is in for a boring career. It's up to you to actively create and seek out opportunities.
There're almost endless possibilities for an Independent character. Some popular choices are explained in more detail here