Acceptance test are a great tool to verify that the application or system works as expected from end to end. Sometimes these tests can be called as end-to-end tests but sometimes end-to-end tests have a different meaning. Another term to describe the same functionality is QA tests and a subset of acceptance tests is often referred as smoke tests.
The idea
The idea is to define a input and the expected output and once the system and all it's dependant services are running the whole system can be verified to work as expected. In a ideal world the acceptance tests would be implemented based on the acceptance criteria of the use case.
The implementation
Acceptance tests can and should be implemented in the code just like unit and integration tests are implemented. The acceptance tests don't neccessarily reside in the same code repository as the code but they can, depending on what's the need.
When a system is API the acceptance tests could be e.g. predefined HTTP requests with predefined responses. These type of tests could easily be implemented with any unit test library or with a help of tools like SoapUI or Postman.
When a system is a web application that's used via browser the tests could be a flow of navigating the application with a browser and verifying that the interactions work as expected. These type of tests can also be easily automated with tools like testcafe and cypress.
Similar tools exist for mobile applications.
Acceptance vs integration tests
Acceptance and integration tests seem very similar and they are. They do basically the same thing but they do it in a different environment. Where a integration test (my idea of a integration test) is executed as part of the systems automated tests in the developers computer and in the CI system the acceptance test is executed against a real system running in a real environment where all the dependant services are running on their own and we as developers aren't neccessarily controlling the system but just executing the tests and observing the behaviour. Acceptance tests can and should also be run automatically by the CI system
When to implement acceptance tests
Just like any other tests acceptance tests can be implemented at any phase of the development process. Sometimes acceptance tests are implemented by someone else than the developer of the system e.g. a QA/tester who's a expert in these type of tests.
One approach is to implement acceptance tests before any code is written. This enables the development of the system with a test driven approach that's called acceptance test driven development i.e. ATDD. Another form of this is called BDD i.e. behaviour driven development.