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.
The effects of wearing a Thinking Cap are added to any other skill-learning bonuses that may be in effect. For example, if a Sim wears a Thinking Cap while teaching a toddler who has drunk Smart Milk, the toddler will learn even faster.
To have a Sim remove the Thinking Cap, select the Sim wearing the cap, click on the cap (not the Sim) and select the option to remove it. It is often helpful to pause the game before doing this.
A Thinking Cap is good for 24 hours of use.