Wikipedia Articles in Regards to Various Types of Exploits
First, there is the Wiki page detailing exactly what an exploit is. This is a very good read and should acquaint anyone with what an exploit is and can be capable of.
Second, there are Wiki pages detailing several types of exploits and detailed information as to how they work. Below, you'll find some great examples of exploits ranging from low-risk web browser information disclosure on up to full system-level compromises.
Some of these methods were even new to me. The Sea-Surf (CSRF - Cross-Site Request Forgery) method is something I had at least considered as possible, but I had no idea that the method had a name and was actively in use in-the-wild.
Some of these methods require some social engineering to trick the end-user (target) into activating the payload; whereas others require no interaction by the target, and they are often unaware that they have been or are being compromised.