Bonus nights earned (and all other designations) as listed on your Activity Page are:
Nights Stayed: Are the paid stays you have made for the year to date.
Bonus nights Earned: is a summary of all the items below it (from Promotional to Rollover)
Promotional: nights earned for a special promotion....for example...a gift birthday night
Rewards Credit Card: are the nights you earn for having the Marriott Rewards card (10 or 15 each year depending on the card you have) and all the additional nights for spending done with the card (your earn 1 elite night for each $3000 charged and paid for on the card)
Rewarding Events: Are special events you do with a hotel (like a function with multiple room bookings on a contract).
Rollover Nights: Are the nights being credited in the current year from a previous year that exceeded the level you attained (example you are Gold Elite and stayed 65 nights...Gold requires 50 nights so the 15 above Gold status rollover to the next year ONLY and give you a head start on the next year's elite nights)
Total Nights This Year: Is just a summary of all the above, including paid stays.
Total Membership Nights: Are the number of paid stays you have made at Marriott in you lifetime in the Rewards program
Hope this helps!