Multiple choice questions can test lots of different abilities and concepts, without taking so much time to answer. Plus, these questions are easy for teachers to grade, as they don't need to spend so much time reading. They are also easy for computers to collect responses and automatically grade.
But they do have some drawbacks. It can be very difficult to create good multiple choice questions. Otherwise, they can be difficult to decipher. Sometimes, the correct answer can be identified by simply looking at the choices, rather than taking the effort of using high-level thinking to find the solution yourself. This gives students who don't know the answer a good shot at guessing the correct answer or copying off of somebody else and getting points they don't deserve.
Short answer questions, as with multiple choice questions, can test lots of skills and can provide even more insight into students' thinking. They give students some room for creativity in their responses, too. Short answer questions are more straightforward than multiple choice questions (no carefully crafted "distractors" in the answers, just the question and a space for the answer), so they are easier for teachers to write and more easily understood by students.
While short answer questions don't take so much time to grade, it is more complicated than with multiple choice questions. Depending on the question, computers may not be able to automatically grade submissions. Teachers need to read each reply to grade short answer test questions. Plus, the creativity students have, usually means that the correct answer can be stated in multiple ways, making it even more difficult to grade.
The open-endedness of an essay allows students to express their knowledge of the answer however they would like. Teachers can more effectively understand what skills a student has and what skills the student may lack. It's not easy for test takers to pretend they know something they don't in an essay, so cheating is not a concern. Plus, long answer questions can be used to improve student writing skills.
Although essays clearly take the cake for deep/high-level cognition, essay questions take a long time for teachers to grade and a long time for students to write. This can be a huge stress factor during tests. Teachers also need to plan essay tests out so that students can finish within a reasonable amount of time. Last, for multiple choice and short answer questions, the answer is either correct or incorrect; however, grading essays could involve opinions and preference.