Calendar Question / event schedule

I have a question about incorprating a calendar into a wordpress site. We are current redoing our company intranet and I am using wordpress for this process. I am looking for an advice for the best way to approach and use the best plug-in to accomplish what I am trying to do.

Currently our companys intranet is hand coded HTML/CSS/Java etc also running MsQL Database on the back side for login information for employeess.

Here is my issue. We have continued education classes that people attend and sign-up for through a calendar that adds their name to that date. This calendar date shows everyone that is attending.

What I need is that when someone clicks on the sign-up button for the event it will add their name to that date on the calendar. What would be the best plug-in or approach to accomplish this?

Thank you for all the help and advice.
Aaron