Right Of First Offer
  
You really like your neighbor's lawnmower. You really like it. Like, sometimes at night, you sneak over to the little window on their garage and peek in at it. And when they're away on vacation...you ride Bruce. Yes, you named him.
You've told your neighbor you want to buy it/him. He's promised you that if he ever sells the lawnmower, you will get first crack at it. He has assured you (many, many times, before he stopped returning your texts) that he won't sell it to someone else until he gives you an opportunity first.
That situation defines the right of first offer.
The term refers to a contractual stipulation whereby one party gets first dips on bidding on an asset. If the second party chooses to sell, they have to give the other party a chance to bid first.