"to suffer or permit to work" is a phrase from the Fair Labor Standards Act, found in Section 203, clause (g). Although seemingly an archaic phrase in a long, technical document, it plays an important part in the working culture of the United States of America.
Standard disclaimer: I am not a lawyer, so the following is obviously not intended as legal advice, but is intended more as a cultural document on expectations around labor in the United States. Please don't try to get legal advice from a dubiously sourced internet site.
The Fair Labor Standards Act is the legal, and in many cases, cultural basis of United States labor law. It was first passed in 1938, as part of Franklin Delano Roosevelt's New Deal Legislation. It has had many things added to it since then, and is now a very long bill that covers everything from what type of safety warning cardboard box balers are required to have, to the requirements for lactation rooms. But in its core, the FLSA focuses on three things: that employees must earn a minimum wage, that there is a maximum amount of hours they can work before they receive overtime, and most importantly, that all work must be compensated.
The last part seems obvious, but it has been an issue of some debate. And it also needs to be defined quite strictly legally, which is why Section 203 (g) states:
(g) “Employ” includes to suffer or permit to work.
This doesn't include just work that an employer mandates or instructs, but any work that the employer allows. Even if an employee volunteers to work for free, an employer has to either disallow it, or, if they fail to do so, to pay for the "volunteered" work. The reasons why this has to be specified is pretty clear, (at least to me, having some experience with employers): it would be pretty easy for companies to only hire workers who, wink wink nudge nudge are so enthusiastic that they will, out of the goodness of their hearts, work "voluntarily". It would be very easy for a manager to start doing things such as leaving a mop outside of an employee's office with the implication that the employee is expected to clean after they are done with their normal workday. By placing the burden on the employer to insure the employee doesn't work, these types of manipulations can be avoided. For the most part. The reader can probably provide many examples of where the letter of this law has been in some way altered.
A few points to put this phrase in context. First, like most things in the FLSA, it applies to "non-exempt" employees, what we would colloquially call "hourly" employees, not to salaried employees. Managers, professionals, and of course rescue workers are not covered by this: an off-duty paramedic can give you CPR without violating this provision.
Secondly, there is a "de minimis" exception to this. Very small amounts of work do not need to be accounted for. If, on leaving work, you empty a wastebasket into a dumpster, that is allowed. Or, if after a shift as a grocery store clerk, you see a stray shopping cart in the parking lot and corral it, that is also below the radar of this law. And, as has come up in recent years, checking e-Mail for a few minutes on a weekend is also allowed.
In the non-profit sector, volunteering for an employer whom one also works for is allowed, as long as the work is not the same. For example, someone with an administrative job at a food bank is allowed to volunteer distributing food, but not allowed to work unpaid hours at their paid job. I am a little less certain about this, especially because in my long experience with non-profits, I have rarely seen it observed.
A developing issue is how this law works with an "information economy" (and yes, I cringe when I type that). Even for non-salaried employees, work is more about knowing things, and less about doing things. Much of the things that an employee studies can be done passively, at home, with no way for the employer to know that the employee is doing so. For example, a salesperson at a hardware store can access their employees website and study what products are available and what prices they are offered at. This directly relates to the employee's job and makes them a better employee, yet it is all done with a public resource, and in a way that the employer has no way of tracking the amount of time involved. And yet, as with earlier examples, this is still something that an employer could "suggest" an employee do. I imagine that there will be (or maybe already has been) a court case to decide where exactly activities like this fit in with the ideas behind the Fair Labor Standards Act, which drew its original significance from the manufacturing economy of the Great Depression.
The text of the act can be found here:
https://www.dol.gov/whd/regs/statutes/FairLaborStandAct.pdf