Acquirer
The acquirer is the bank that holds the merchant's account and takes on the risk that the merchant will not deliver. It is the merchant's side of the card network.
Issuer
The issuer is the bank that gave the cardholder the card. It decides whether a payment is approved, and it lends the money in the credit case.
Card networks
A card network is a rulebook and a switch. Visa and Mastercard do not lend anyone money, do not hold anyone's account, and do not decide whether your payment is approved. They define what the messages mean, move them between banks, price the transaction, and settle the net positions at the end of the day.
Processors, gateways, and PSPs
Between you and your acquirer sits a stack of companies with overlapping names. The words get used interchangeably, including by the companies themselves, and the confusion is not harmless: when a payment breaks, the name tells you who can actually fix it.