When you're on the job, you should be paid for every second that you're working. There is no such thing as a "free lunch," for you or your employer, so there's no reason not to get paid when you work.
An employee who chooses to stay late should still be paid, even if the employer does not want to do so. In the future, the employer might limit the employee's hours, but they can't ignore that the employee did do work.
How much should you be paid if you work more than 40 hours?
If you work more than 40 hours in one week, the Fair Labor Standards Act requires most employers to pay overtime. Overtime is usually at least one and one-half times the normal rate of hourly pay. Some people are exempt from these rules, like some commission-based sales and farm workers. It is up to you to find out if overtime is limited in your industry.
Working off the clock: Is it legal?
Working off the clock is not something anyone should do. During that time, you're not being compensated for the work, yet your employer still benefits from your work. An employer is required to pay you if you're working, regardless of how much time you've spent on the job that week. There are cases where employees may choose to work to help their coworkers or the business and not expect payment, but these situations are taken on a case-by-case basis.
It's important for every person to know how they'll be paid when they work. If you're on the job, you deserve compensation. An attorney can help you seek compensation from your employer for the hours you worked.