Is it possible to add some code that will tweet a message (or would offer to post a tweet) once a user submit a form yo post a message on my site.
Do I need to point it to my tweet account? Just trying the process.
I use PHP and jQuery on my site.
Yes it is possible. Yes, you do need to point it to some twitter account in your control (you could create a new one for your app/website). See the twitter php libs.