Issue spotter exams are those where the professor's primary focus is for the student to identify the legal issue presented by the facts and discuss it. Many times issue spotter exams are race horse meaning that there are a boatload of issues and you can't possibly address all of them in the time alloted. It tests you on how quickly you can analyze the fact pattern and identify issues. The goal there is to get the big fish issues first and then work your way down to smaller issues as time permits. Usually on issue spotter/race horse exams it's more important to identify more issues than do thorough discussion of each issue.
There are also plain issue spotter exams where you have plenty of time to spot and discuss all of the issues -- I like those. The call of the question is usually "discuss the rights and liabilities of the parties." Here, it's more important to thoroughly discuss the issue. Also, there may be hidden issues that only the most prepared students will identify.
Thinker exams are those were you have plenty of time, but you're presented with very novel issues. Things that are several steps beyond what was discussed in class. If you know the material you can take those steps, but it takes you time to do it in the exam. Thinker exams may also ask you policy-type questions. For example, if the legislature were to adopt XYZ statute, discuss the economic and legal impact such a statute would have on ABC. Or a professor may ask you to draft a statute to address a particular legal problem where there are good arguments on each side of the issue.
Also many professors will give multiple choice exams.
Key is to figure out what type of exam you'll be getting (talk to former students and ask the professor) and keep that in mind as you study and prepare your outlines.