A rollover is between different "qualified retirement accounts" of the same tax treatment. It can be a Traditional 401(k) to a Traditional IRA, or one Traditional IRA to another Traditional IRA. Or from a Roth IRA to another Roth IRA. Or a Roth 401(k) to a Roth IRA.
There are no taxes due for a rollover.
A "conversion" moves funds into a Roth account from a non-Roth account.
At the time of conversion (in the same tax year), you have to pay taxes on any funds that were untaxed (contributions previously deducted from your taxable income, as well as gains that haven't been taxed yet). Yes, you can spread these over multiple tax years by doing the conversion in small batches.
A "backdoor Roth contribution" first makes a contribution into a Traditional IRA, chooses not to take a tax deduction (so you avoid the requirements around deductions), and then quickly rolls the Traditional IRA into a Roth. Works great if you don't have other money in the Traditional IRA. If you do have untaxed money in Traditional IRA accounts, the problem is that the conversion draws proportionally from the untaxed money and the basis. One way to avoid this is to remove the untaxed money from your IRA by doing a "reverse" rollover into a Traditional 401(k). Just holding multiple IRA accounts does NOT avoid the problem.