Qualified Distribution
  
A Roth IRA allows you to put money away for retirement, with the government offering you tax advantages along the way. However, there are rules about taking money out. You can't just decide at age 45 that you want to open a surf shop at Teahupo'o, clean out your Roth IRA, and use the money to purchase your inventory of wax and sunglasses. If you take money out early, you face a penalty.
A qualified distribution is a withdrawal from a Roth IRA that follows all the rules. You get the money without added taxes and without a penalty.
There are two basic forms of IRA (individual retirement account). One is the traditional IRA. In this version, you contribute pre-tax dollars...meaning you get a tax break on any cash you put into it. The money you put into the account (hopefully) grows over time. When you get to retirement age, you start taking the money out. At that point, your withdrawals get taxed as ordinary income.
The other version is the Roth IRA. In this version, you contribute after-tax dollars. You pay your taxes on the money when you earn it, then put the money into the Roth IRA. No tax breaks at that point. Then the money bakes for a few decades in the Roth IRA, (hopefully) growing over time. Now, it's retirement time. You start taking the money out. For the Roth IRA, you don't pay taxes on the money you take out.
So for the traditional IRA, the tax break comes at the time of contribution. For Roth IRA, the tax break comes at the time of withdrawal. The IRS gets out the microscope for those Roth IRA withdrawals. The tax people want to make sure you follow all the rules when taking your money out, since you're going to get the cash without paying any taxes on it. That's what makes the distribution qualified. It fits all the rules. It qualifies for that favorable tax treatment.