Let's see if I can get it right this time:
Your html form seems to be in order...
To get your values to the PHP file use $POST['login_name'] and $POST['login_pswd'].
If the user is suppose to be "logged in", then you might need to work with $_SESSION's.
Regarding the PHP2MySQL side: I do not work with MySQL so I can't help you, sorry.
Try http://www.w3schools.com/sql/default.asp, they have usually have good tutorials.
I guess that all that you need to do is "iterate" through your database until a match is found. I would like to point out that I'm just guessing.
Also, I do not know anything about security issues regarding user management.
But I hope this helps you "a little bit".
Good luck.