Authorisation of a transaction is another name for payment authorization, the step in which a payment is approved before it is carried out. The British spelling is common in the UK, the EU and other countries that follow British usage, and it appears in legislation such as the EU's payment services directives. The main entry covers how approval works in card and account payments.
The payments-law sense
In payments law the phrase has a precise meaning. Under the Second Payment Services Directive (PSD2) and the UK's Payment Services Regulations 2017, a payment transaction counts as authorised only if the payer has given consent to it, in the form and by the procedure agreed with the payment service provider. Consent is normally given before the transaction or, if agreed, afterwards, and it can be withdrawn up to the point at which the payment becomes irrevocable.
The distinction matters for liability. If a transaction was not authorised, the provider must generally refund the payer promptly, subject to limited exceptions such as fraud or gross negligence by the payer; see unauthorised transactions. Authorisation is also separate from authentication: strong customer authentication checks who is giving the instruction, while authorisation is about whether consent was given. A payment the customer was tricked into making, as in authorised push payment fraud, is still authorised in law, which is why the UK introduced separate reimbursement rules for it.