Does "regular" contributions and distributions mean amounts I contributed or withdrew on a regular basis? Or does it mean they were not "conversions?"

I can only think of the 1099-R entries where it asks if these were regular or sporadic payments received. 

For example, a retirement plan would be regular monthly payments whereas an IRA withdrawal could be sporadic. 

When entering a conversion, select not regular, and continue answering the questions about what you did with the money, like rollover.

See 1099-R codes. See Table 1. Guide to Distribution Codes. 

If this does not help, please reply with more information.

The answer depends on the context.

"Regular" contributions are those deposits into your retirement account that are not deposits of transfers, rollovers, conversions or recharacterizations.  Regular contributions are ordinary contributions.

Similarly, ordinary distributions would generally be those other than recharacterizations (reported with code R or code N), returns of contributions (reported with code 8 or code P), or distributions made under the Employee Plans Compliance Resolution System (EPCRS) (reported with code E).

However, when upon entering a Form 1099-R TurboTax asks if you received "regular" payments, TurboTax is asking if you received periodic distributions in the form of an annuity (pension payments would be such payments).  TurboTax asks this in case the taxable amount reported in box 2a of the Form 1099-R is incorrect, you made after-tax investments in the plan, and you need to use the Simplified Method to calculate the taxable amount.