Credit cards are usually issued by a card issuer (bank) and are co-branded with one of the card associations. For example, the Sapphire card is a particular credit card issued by Chase Bank, but is a Visa card.
American Express (Amex) is a little bit weird in that they act as both the issuer and the association for many of their cards. (They are less widely accepted than Visa or Mastercard).
See this American guide to choosing a credit card[1]. Factors that people consider include interest rate, amount of signing bonus, rewards (such as a percentage cash back on each purchase or a percentage of frequent flyer miles on each purchase), yearly fee to use the card, and other services that come with membership such as warranty protection, concierge, discounts, and reserved tickets for events.
What cards are available to a person depends highly on their credit history and income.