An achievement test is designed to measure what a student has learned, not her innate intelligence or problem-solving skills. The tests schools administer to students are achievement tests, which is why these tests sometimes play a role in school funding or employment decisions. School administrators and government officials reason that, if students are performing poorly on a test of what they've learned, their teachers must not be effectively teaching them. The ACT, one of two college admissions tests, is also an achievement test that measures learning in math, science, reading and writing.
An aptitude test is designed to test a student's underlying intelligence, ability to learn and ability to reason. These tests frequently phrase their questions in such a way that students are required to use logic and think quickly to get the right answer. The SAT, a popular college admissions test, is an aptitude test because it measures students' ability to reason their way through questions rather than just recite knowledge they've learned in school.
An IQ test, like an aptitude test, measures a person's aptitude for a specific subject. The score, however, is scaled in such a way that 100 is always an average score. Originally used to determine how students were developing relative to their peers, IQ tests are now routinely given to adults. The most well-known IQ test is the Stanford-Binet Intelligence Scale, and the test has been used for decades to measure intelligence in a wide variety of contexts.
There's no clear line between achievement, IQ and aptitude tests. Aptitude tests are supposed to measure innate aptitude, but can't fully do so. After all, a person can't even read the questions if she's not taught to read. Similarly, an achievement test may be measuring a combination of learning and aptitude. IQ test scores are not immutable, as one might expect if they were measuring pure aptitude. Instead, scores tend to increase with education and tutoring.