Forget about the DataGrid. A DataGrid is for display purposes and has nothing to do with saving data to a database. You retrieve data from a database into a DataTable using a DataAdapter and its Fill method. You can then do whatever you like with that DataTable, including bind it to a DataGrid or other controls. The user can then edit the data to their heart's content. You then save any changes made in the DataTable using the same DataAdapter and its Update method. The DataAdapter doesn't care where the changes came from, so the DataGrid is irrelevant to the process of saving the data.
I suggest that you do some reading on ADO.NET, which is the standard .NET data access technology. Fellow member TechGnome has a link to his own tutorials in his signature. The tutorial links in my signature will also shed some light on the topic, as will many other tutorials on the Web.