What is a Roth 401k or Roth 403b? Is it a new type of plan?

Like the Roth IRA is to the traditional IRA, a Roth 401k is to the traditional 401k. You contribute after-tax dollars, but then later get the money out out tax-free.
No, it is not a new type of plan. Designated Roth contributions are a new type of contribution that can be accepted by new or existing 401k or 403b plans. This feature is permitted under a Code section added by the Economic Growth and Tax Relief Reconciliation Act of 2001 (EGTRRA), effective for years beginning on or after January 1, 2006. If a plan adopts this feature, employees can designate some or all of their elective contributions as designated Roth contributions, (which are included in gross income) rather than traditional, pre-tax elective contributions. So, starting in 2006, elective contributions come in two types: traditional, pre-tax elective contributions (elective contributions are also referred to as elective deferrals) and designated Roth contributions.