Before diving in, here are a few key terms to help you navigate the payment industry. Please note that these definitions are intentionally simplified, our goal is clarity, not exhaustiveness.

• Issuer: This is the bank that issued your client's credit card. When your client pulls out their Visa card from BNP Paribas, BNP is the issuer. They're the ones who say "yes" or "no" when you request to charge the card. The issuer is also the one who'll give you headaches with sometimes overly strict anti-fraud rules that block legitimate transactions. They are taking a commission on every transaction : interchange fees
• Acquirer: This is indeed the bank that allows you to process credit cards on their behalf, but concretely they're the ones taking the risk. If your client (who pays on your site) does a chargeback 6 months later, it's the acquirer who will come after you. They give businesses authorization to process on their behalf but in exchange they monitor your transactions and can cut you off overnight if they find your business too risky. To do that they take a small commission (acquirer fees)
• Processing Network: Basically, when your client pays on your site, the money goes from the client's account (the issuer bank account) to the business account (acquirer bank account), but you can imagine that issuers and acquirers don't all know each other, so there's a middleman between them: the processing networks. It's the highway between the acquirer and the issuer. You know the biggest ones: Visa and Mastercard. So when you pay, your card info goes through their network to ask the issuer "does this guy have enough money?" They take a small commission along the way (scheme fees) and dictate the rules of the game.
• Payment Service Provider (PSP): This is the guy who simplifies your life by handling all the technical mess. Instead of negotiating with 10 different acquirers, you go through Stripe, PayPal, Adyen... They already have the connections and offer you a simple API. In exchange, they take their margin on each transaction (often 1-10%).
• ISO: Independent Sales Organization, this is the salesperson who goes prospecting merchants for acquirers. Imagine a sales rep selling payment terminals to restaurants. The ISO isn't a bank but resells an acquirer's services while taking a commission. They're often the ones offering you "super deals" with hidden fees.
• Merchant of Record: This is the legal entity that appears on your client's bank statement and is responsible for the transaction. If you sell via a marketplace, it's often the marketplace that's the MoR, not you. It's convenient because they handle taxes, chargebacks, compliance... but are legally absorbing every risks
• Chargeback: When you pay with your credit card, you have protection rights. If the merchant doesn't deliver, scams you, etc., you can initiate a chargeback (bank dispute). You often have up to 3 to 6 months to do this depending on jurisdictions.
• Acceptance Rate: Due to numerous factors (compliance, fraud, technical issues...), when you pay by card sometimes it can end in... error. And that's problematic because a large percentage of the time the customer will leave and won't retry after this first error. The percentage of transactions that go through on average without error is the acceptance rate. The acceptance rate is higher in local corridors (European client buying from a European business or US client buying from a US merchant...).
• Tax Management: When you're a business wanting to sell in a country, you often need to collect VAT (or Sales Tax in US/Canada/...) and remit it to the government/state according to a precise schedule. It's very annoying because it's a lot of paperwork and especially because you often need to create a local entity.
• Payin: This is money coming in. When a client pays by card at a business, it's a payin. Money goes from the client's account to your PayPal account, for example. But once it's on PayPal, you need to retrieve it to your own account and that's a... payout.
• Payout: A payout is money leaving your payment processor (payment service provider, acquirer,...) to your bank account.