It can be compared with proficiency tests, which measure a learner's level of language, diagnostic tests, which identify areas learners need to work on, and a prognostic test, which tries to predict a learner's ability to complete a course or take an exam.
Example
The learners have finished units 1 - 4 of a coursebook and the teacher now gives them an achievement test based on what they have seen in these units. The test is taken from the teacher's book.
In the classroom
Achievement tests can have many additional functions aside from evaluation. Learners can for example develop an action plan for further study based on the results of an achievement test.
Further links:
https://www.teachingenglish.org.uk/article/test-question-types
https://www.teachingenglish.org.uk/article/testing-assessment
https://www.teachingenglish.org.uk/article/young-learner-testing