Reasoning is the ability to come up with valid conclusions based on given premises. Logic is the science that tries to formalize reasoning, and it is a branch of mathematics.
Probably every human being is able to think reasonably to some degree. However, people tend to make mistakes from time to time. Some people seem to be talented, they arrive to wrong conclusions less often than others. In general, however, reasoning ability seems to be something that can be learned and improved by training.
I studied formal logic as a part of my Master's degree in Computational Intelligence. Formal logic helps translate thoughts into a language that can be understood by other people and even machines. A branch of computer science called automated reasoning deals with the solving of logical problems by computer programs. This enables to verify conclusions made by human beings, including mathematical proofs, and may even have computers solve problems on their own.
Reasoning ability is probably one of the most important requisites to become an intellectual since intellectuals are expected to back up their statements with arguments. However, intellect is more than just reasoning ability, it is also composed of knowledge and the ability to notice patterns and come up with new ideas.
A couple of years ago I developed a logic quiz based on set theory that may help you discover whether you have the special talent to draw correct logical conclusions without making too many mistakes. It might help you gain a more realistic view of yourself, your strengths and weaknesses. The quiz has been taken by several thousand Internet users by now, which enabled me to create statistics, so you will not only see how many mistakes you made but also how you compare to other users.
Link to the Logic Quiz - Set Theory
Contact: cdvolko (at) gmail (dot) com