After-Tax Contribution
  
When you put money into a retirement account, it matters whether the funds come before taxes or after taxes. An after-tax contribution means you have already paid taxes on the money, i.e. the funds have been counted for income tax purposes in the year you received them. When you take that money out, decades later, it won't be taxed. That is, when you put after-tax funds into retirement, you don't have to pay taxes again down the road when you use them in retirement (though you might have to pay taxes on any investment gains you make).
An alternative world of deferred tax savings exists, essentially called a pre-tax investment. These tax-deferred savings vehicles include things like 401(k)s and IRAs which come in a range of flavors. These forms of investments allow you to avoid taxes now, though you will have to pay taxes on them when you take money out during your retirement years. The benefit is that you have more money to invest, because the government hasn't taken a bite out of your wallet yet. As this grows over time, the extra funds should multiply, leaving you better off in the end, even though you have to eventually pay the taxes down the road.
The rationale is that, in your peak earnings years, you're paying high marginal tax rates...maybe 30-35-40 percent. When you retire, you'll likely be paying a lower tax rate as you withdraw (and then pay taxes on) your IRA and 401k savings vehicles.
The big idea here: When you make an after-tax contribution to a savings for retirement vehicle, the tax is thus already paid. You don't pay it again.
Wasn't that a line from Casablanca? "Pay it again, Sam"...or something like that?