Prime Offers appear at the top of the screen and pop up in windows every time you start the game. Prime Offers give you the chance to receive game items in exchange for purchasing or using gold during one of the offers.
Prime Time, Prime Day and Hot Deals are the types of offers that reward you for purchasing gold should you meet the conditions of the particular event on offer at that time. Prime Spending rewards you for spending gold.
Prime Sale allows you to buy handbooks from the Library for 10% less gold than normal. 1 handbook is offered at a discount each time Prime Sale runs. You will have to wait for the handbook you want to be offered if you want the 10% discount.