Adding an apostille to a document is often referred to as the attestation. It may also be called, certification or getting a document apostilled.
The most common, and arguably correct term, is legalisation. Documents are legalised for use in another country when the apostille has been attached.
For clarity, attestation is by definition the confirmation of correctness. It also refers to the witnessing of an action or event. It is therefore easy to see why the apostille service may be referred to as attestation.