It depends on your preferences. If you want a credit card issued by Discover then you should have good-to-excellent credit history. To get approved for Discover you should show high credit score, regular payment history, and established credit. The exception is student credit cards. If you are a student you can get a Discover credit card for Students with fair credit history.
But if you want a credit card serviced by Discover Payment Network and issued by any other bank, you will be able to find a card that fits bad credit history. For example, Cerulean Discover® credit card is issued by Services Credit Union and has a Discover label on it.