In a nutshell, envelope budgeting means that you have cash on hand before spending money. In the old days you would cash your paycheck into actual cash, and then divvy up the cash into envelopes marked "rent" or "gas" or "clothes" or what have you. When it came time to buy something or pay a bill, you could only spend the money that you had in that envelope. There's nothing keeping you from moving money from one envelope to another, but the assumption is that you're wise and you're not going to take all of the money out of "rent" to go buy stuff.
Obviously we don't use cash and envelopes anymore, but the theory is still there. A financial management system that supports envelope budgeting wouldn't show you what your checking account balance is as much as how much money you've got in your set of virtual envelopes.
This is different than what Quicken or mint.com or a bunch of other financial management tools do. Sure, Quicken lets you categorize your expenses, but it's more forensic accounting ("where did my money go?") than a planning/saving tool. The power of envelope budgeting is that you're giving each dollar a name and a purpose, putting you in control of your money. Envelope budgeting is also much better than traditional budgets for more occasional expenses (property taxes, car repairs). I know I've got to pay property taxes in October, and I'm putting a little money in that envelope now with each paycheck, so that when that big bill comes along, I'm all set. It also works very well for those with irregular incomes -- you can prioritize your expenses, funding the most important envelopes first.
There are a variety of online services and dedicated programs that support the envelope method. I tried out mvelopes.com a while back, but it's expensive and they want to know your banking login information, which I'm a little creeped out about.
I'm taking a look at neobudget.com now -- look for a post in the next week or so.