I am here in US on an M 1 visa and my fiance is a US citizen. Can you please explain the process of court marriage in our case? Thanks

You need to look up the procedures based upon the county where you live. Each county's procedures vary. Most county's publish information about the process as well as documents needed to get married on their websites. Once married, you can start the green card process. You can read more about the green card process at http://myattorneyusa.com/adjustment-of-immigration-status.