On a co-branded card, the brand of a company other than American Express will be displayed prominently on the card in addition to the American Express payment network logo and whatever other American Express branding.
Credit underwriting standards for an issuer's co-branded cards are believed to be less strict than for American Express issued non-cobranded products.
However, you can still certainly be denied for an American Express co-branded product.
A non-exhaustive list of co-branded cards issued by American Express in the United States are the cards it issues in its Delta Air Lines and Hilton Hotels product families.
For co-branded cards, signup bonuses and rewards will usually be in units of the co-brand's loyalty program.
Significant exceptions are the co-branded American Express Platinum charge cards such as the American Express Charles Schwab Platinum charge card and the American Express Morgan Stanley Platinum charge card.
In the United States, for the American Express Platinum charge card, the signup bonuses and rewards are in units of Membership Rewards regardless of whether the card has a co-brand.
American Express has also used the term "Platinum" to apply to one its personal Delta co-branded credit cards and one of its small business Delta co-branded credit cards.
Other than having the word "Platinum" in the name, the Platinum charge cards and the Delta co-branded credit cards are unrelated products with unrelated annual fees and benefits.