Note on converted Roths--The 5 year rule pertains to contributions and earnings.Let's start with the person who opens a Roth and makes periodic contributions. That person can withdraw those original contributions anytime with no tax or penalty. The five-year clock for earnings on those contributions starts Jan. 1 of the year for which your first Roth contribution was made, and it doesn't reset each time you make a contribution or open another Roth.
As for the person who converts to a Roth: In a conversion, you have to hold the assets in a Roth for five years or until turning 59½, whichever comes first, to make penalty-free withdrawals of your converted amounts. Here, each conversion has its own five-year clock.