The assignment content describes three use cases for a health club membership process. The first use case, 'Process New Customer', handles the registration of new customers and calculates their payment amount based on the desired membership length. The second use case, 'Process Payment', handles various payment methods such as cash or credit card. The third use case, 'Process Declined Credit Card', handles situations where a customer's credit card is declined. Additionally, there is another use case, 'Issue Membership Card', which issues a membership card to the new member.