Many security scanners like nikto, nessus, nmap, and w3af sometimes show that certain HTTP Methods like HEAD, GET, POST, PUT, DELETE, TRACE, OPTIONS, CONNECT, etc are vulnerable to attack.

What do these methods do and how can they be exploited?

I'm looking something more creative than common exploits like POST or GET injections (e.g., changed fields). It would help me to understand if your answer showed me a brief example of the normal usage of the header as compared to an exploit technique of a header.