Unemployment insurance is designed to provide you money between jobs when you lose your job through no fault of your own, but is there ever a case where you can receive unemployment payments even when you resigned? Yes there are. There are not very many (and they vary by state--remember unemployment is a state decision, not a federal one), but they do exist.
If you're contemplating quitting your job, and wonder if you could possibly get unemployment payments, here are four times when it's possible.
Valid personal reason
If your wife is in the military and is transferred from Florida to Idaho, and you quit your job to follow her, you'll be eligible for unemployment payments from Florida. (You always get the payment from the state you worked in.) If you live in California and quit your job to follow your spouse for any job, you can be eligible there--but in Florida, it has to be a military job that triggered the relocation. Each state has their own rules, but here is a list of which states offer unemployment for trailing spouses.
Constructive discharge is where your job is so horrible that any rational person would quit the job. Lots of people think this is they key to getting unemployment after quitting a job, but the standards are quite high. It's not as simple as "my boss was a micro-managing jerk."
Generally, your situation has to be both miserable and illegal. (It varies by state, of course). So, you may be eligible if your boss was sexually harassing you, or you may be eligible if there were health and safety violations that put you in danger. If your situation is just annoying or difficult, you probably won't qualify.
If your company wants you to move across the country and you don't want to move, you'll be eligible for unemployment if you don't take the new job. If they want you to move across town, it's a little more iffy. It will all depend on the state and the reasonableness of the commute. This reasonableness changes a lot with states. In Utah, where the speed limit is 80 miles per hour on parts of the interstate, a 50-mile move might be considered a reasonable commute. But, if that 50 miles is in a hugely populated area with heavy traffic where 50 miles can take 3 hours, it might not be reasonable, in which case you'd be eligible for unemployment.
You're called in and told that you have to resign or else. (Or else what? They'll fire you?) If the resignation is their idea, even if you sign a letter, you should still be eligible for unemployment. The key thing here, though, is to document everything clearly and get it in writing. Many companies that want you to resign also don't want to be dinged for unemployment so will fight it. Keep copies of all communications and be prepared to appeal at the unemployment office.
Remember each state is different and has slightly different rules, so don't take this as the absolute truth for your situation, but these are general guidelines. When in doubt, call up your local department of labor office or local unemployment board. Additionally, if you're denied, always appeal. You have nothing to lose by appealing.