IQ is measured by an intelligence test. One common set of tests is the Wechsler scales, including measures for preschoolers, children and adults. Aptitude is measured by a wide variety of tests, such as the Scholastic Aptitude Test, which is intended to measure readiness for college-level material.
The first intelligence tests were devised by Alfred Binet, and were intended to distinguish "normal" and "inferior" children. The term "intelligence quotient" refers to dividing mental age by chronological age, but this is no longer how IQ is measured. The idea of mental age has largely been abandoned. Today, IQ scores are based on factor analysis of responses to a variety of types of tasks.
Aptitude tests have a less easily defined history, because there are so many of them. These include aptitude tests for various careers and for various levels of schooling.
Both IQ tests and aptitude tests are controversial. There is debate about whether intelligence is a single trait or multiple traits, and about how those traits can be measured. Aptitude tests are somewhat less controversial, in part because it is clearer what they are supposed to measure. For example, the SAT is intended to measure readiness for college; if people with high SAT scores tend to get better grades than people with lower SAT scores, that is some validation of the test.