Once you've appropriately researched your discount-broker alternatives and decided on which one is right for you, you'll probably want to go ahead and open the account (rather than simply resting on your laurels for having done your homework).
Opening an account doesn't require much more than filling out an application. Don't worry about the word "application" -- we haven't heard of anybody getting turned down. However, being approved to open a margin account (i.e., an account that permits you to borrow money from the broker) might be a problem if you have a horrible credit history. But an account that is funded entirely by your own cash is most likely going to be approved.
You download the application forms, sign them, enclose them in an envelope with a check to initially fund your account, and you'll receive confirmation of your ability to start trading in pretty short order. Voila!
The application will want you to choose among different types of accounts, which we explain here.
And to start investing, you don't need much more than your computer, some moolah, and a bit of investing know-how.