Cognitive tests assess a variety of capabilities, including linguistic aptitude, executive function, spatial reasoning, short-term memory, self-assessment, personality facets, among other qualities. The most famous type of cognitive assessment is the IQ test.
IQ tests are supposed to correlate with a general intelligence factor, but it has not been proven conclusively. The most commonly accepted IQ measurement is Raven's Matrices, which is a visual measure of "meaning making."
Achievement tests also assess a variety of things, but they focus on skills and knowledge learned over a period of time, typically in a classroom setting. The scores from an achievement usually result in a grade or certificate of advancement, or result in admission to a program of some kind. The most iconic achievement tests are standardized tests such as the SAT and ACT.
Standardized tests are generally given on a large scale so that large groups of examinees may be compared for evaluation purposes.