Roth 401(k)
  
The more common "Roth" distinction comes with IRAs. There's a traditional IRA...and there's a Roth IRA.
The main difference is this: you put pre-tax dollars into the traditional IRA, and after-tax dollars into the Roth IRA. You get immediate tax incentives to put money into a traditional IRA. Meanwhile, you get tax breaks down the road with a Roth IRA, avoiding taxes when you take funds out for retirement.
Not surprisingly, the same distinction comes into play with a Roth 401(k). The regular 401(k), the kind you probably think of most of the time (at least, the one you think of on the rare occasions when you're actually thinking about these things), gets filled with pre-tax dollars. You receive a tax break now to contribute to the fund, but you have to pay taxes down the line, when you make withdrawals.
The Roth 401(k) involves after-tax contributions. So...you pay the taxes on that money now. But, come retirement time, you can withdraw the funds tax-free. A little gift to Future You.