Adyen, Stripe, and PayPal are the most recognizable payment service providers. A PSP lets a business accept payments without holding its own merchant account or signing acquirer contracts, bundling the gateway, processing, and often acquiring with merchant onboarding, risk screening, and settlement reporting.
Becoming a payment processor in your own right works the other way, since it requires acquirer sponsorship, card network registration, PCI DSS validation, and your own settlement and ledger stack. Platforms usually start building their own payment software at that point.