I own a small start up manufacturing company in California. We are on a tight budget and cannot afford to pay overtime. I do not force my employees to work overtime however we are very busy and my employees have asked if they could work extra hours at regular pay rather than I hire more employees. Can I allow them to work extra hours at regular pay as long as they sign a written agreement to do so?

No you cannot. CA law does not allow the employee to waive his rights to overtime pay. Here is an explanation in laymen's terms of the overtime laws in California. See especially Question 10.
http://www.dir.ca.gov/dlse/FAQ_Overtime.htm

As a matter of fact, neither does Federal law. Under Federal law, an employee who works overtime has to be paid at overtime rates - no exceptions. What is considered overtime varies with the state, but not the law.

There was a lot of discussion not too long ago about amending the law to say that an employee who wished to take comp time instead of overtime could do so, but it was never signed into law.

Is that in California that if the employee works more than 8 hours each day, it is overtime?
While in some other states, only when the employee works more than 40 hours each week (not matter how many hours they work each day), it is overtime~

That is correct. If you will read the page I gave you the link to, you will see the requirements for California, which is one of the few states that has a daily overtime requirement.