Just to add to what Mishika said, you can provide a form in your web site with "username" and "password" fields. Then use that information to login with web services or REST (web services is simpler in my opinion if you are using PHP).
There a couple of options at this point:
Option 2: Manually create the first DynaForm in your web page and when it is submitted, use web services or REST to automatically start a new case which will be started with the logged in user.
Option 3. Append the session ID to the URL to start a new case in an iframe:
http://wiki.processmaker.com/3.0/Proces ... atic_Login