PHP Login Script With Remember Me Option HTML Form Username Password

i read your article on creating a login script with php, but i was wondering if you can take it a step further, for example, i would like to have a login script where it asks for the username and password and then there's an option also included that asked the visitor for an option that says "remember me" like this for example:

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User Name		Remember Me?
Password

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this is exactly what i want to put on my html form using a login form for my web site. i've see this similar to form forums and blogs that have this feature on thier websites and i alwaays wondered how they do it. i think there's must be some kind of cookie and i know for sure it involve a database, which is no problem for me because i know mysql. if you guys can show me how to do this in a very simple way that would be great. i've found some other tutorials but they are way complacated and the more harder it gets, i want something simple, a script with code i can understand, you know, for a beginer like me.