The Thinking Cap is an aspiration reward that was added in The Sims 2. It is also known as the "Thinking Person's Silly-Hat-With-A-Light bulb", which is a fairly good description of it. Silly or not, it works.
If a Sim in gold or platinum aspiration puts it on, it lights up, and the Sim will learn faster while wearing it. It will continue to work even if the Sim drops into green aspiration while wearing it. However, if a Sim in green or red aspiration puts it on, it may fail. When it fails, it sputters, and the Sim will pass out and lose a skill point.
A Thinking Cap is good for 24 hours of use.
It is rumored that if a teen wears the thinking cap while doing homework, the wearer will do homework much faster than not wearing it.
Also, if worn while studying from the Books First for Learning: A Bookshelf of Education, the wearer will gain skills very fast, a benefit for knowledge sims. This is because anyone studying from the Books First for Learning: A Bookshelf of Education learn 500% faster.