Aptitude tests measure the ability to learn. More specifically, these tests determine the ability to reason, solve problems, understand language, and make deductions. The key word here is ability. We can even say potentiality. What test creators are aiming for is whether the test taker has the capabilities to succeed in a given environment (school, company, or something else). To be sure, there are many critics out there that say that these tests are not reliable, but the general public still uses them as a gauge of ability.
Achievement tests differ in a fundamental way. Achievement tests measure the extent to which a person has mastered a given field. For instance, it can be a subject matter like biology or a field like finance. Usually preparation of study is expected and required.
As for examples, the SATs, GREs, and LSATs can be viewed as aptitude tests. They seek to determine whether a person has the mental ability to succeed in a given institution. So, for instance, it is a well-known fact that many law school only look at numbers. Subject tests, such as college bound Achievement tests are an example of achievement tests (hence, the name). We can also categorize the Bar exam and Medical School Step 1, 2 and 3 exams in this category, because they measure how well a person has mastered a body of literature.
Finally, it should be stated (as I intimated above) that all of these standardized tests are not without critics. There are plenty of them. They say that these tests are biased and imperfectly measure intelligence and have little correlation with success.