"Elite Nights" or "Elite Qualifying Nights" or "EQN" are nights counted towards Elite membership in Marriott Rewards. There are 3 tiers of membership, Silver, Gold Platinum. See Elite Membership Overview
Marriott has many different promotions, do you recall which one?
"5,000 Bonus Points and 5 Elite nights after your second stay is our way of thanking your for your loyalty to Marriott Rewards® — from March 1 through May 31, 2010."
That's an excellent offer. It's also targeted, payment with the Marriott Rewards Credit Card is required. Some received a very enticing offer - 10,000 Bonus Points and 10 Elite nights after the third stay offer. See FlyerTalk