Unemployment benefits are intended to lend a hand to people who have lost their jobs through no fault of their own. If you resign from your job, you're making a choice to leave, which generally makes you ineligible for benefits. However, there are some circumstances under which even a person who quits a job would be allowed to receive jobless benefits.
If you hope to receive unemployment benefits after you quit your job, a lot will ride on your motivation for quitting. Eligibility for jobless benefits is determined by state laws rather than federal law, so there is some variation. But the general rule is that if any reasonable person could have been expected to leave their job when faced with the same circumstances as you, then you have a legitimate claim to benefits. The law calls this "good cause," and it means that you left because your working conditions had become intolerable in one way or another. Note that a "good reason" for quitting isn't necessarily the same thing as "good cause." Your work may be mind-numbingly boring, and that's a perfectly good reason to seek other opportunities, but it's not good cause as far as jobless benefits are concerned.
To show good cause, you'll have to demonstrate that you would have suffered harm -- physical, financial or otherwise -- had you remained on the job. For example, if your employer severely cuts your pay or hours, you may no longer be able to support your family on your wages. In such a case, you'd probably have a legitimate case for benefits. Other common examples of good cause include an unsafe workplace, harassment and illegal discrimination.
General job dissatisfaction isn't good cause for resigning. Your boss may be annoying, but unless he is creating a hostile workplace with harassment or discrimination, you can't claim benefits if you quit. The same applies if you decide to leave because your prospects for advancement are limited, you were passed over for a promotion or your request for a raise was turned down. You may be unhappy, but your job situation hasn't changed materially.
If your employer gave you the choice to quit or be fired, then your resignation might be for good cause. In that case, it would depend on the reasons behind the ultimatum. If you got caught drinking on the job, stealing money or sexually harassing a co-worker, then your claim for benefits will probably be denied because you lost your job through your own fault. But if your company was undergoing a general "housecleaning" and your boss wanted to give you a chance to resign because it might look better on your resume, then you have a good chance at getting benefits. In advice it gives to federal political appointees -- who are expected to resign when a new American president takes over -- the Department of Justice points out that it's critical in these types of situations to wait until you're actually asked to resign. Otherwise, your resignation may be considered voluntary, making you ineligible for benefits.
According to the Nolo legal reference site, some states -- but not all -- allow you to claim benefits if you were forced to resign for compelling personal reasons, such as an emergency that requires you to be at home. Check with your state's unemployment insurance agency to determine whether it makes such allowances.