Say I have a website and a database of that website hosted locally on my computer (for development) and another database hosted (for production)...ie first I do the changes
We do this in the (Ruby on) Rails world by writing "migrations," which capture the changes you make to the DB structure at each point. These are run with a migration tool (a task for rake), which also writes to a DB table so it knows whether a particular migration has been run or not.
You could make a structure like this for your dev platform (.Net?), but I think that in other answers to this question people will suggest available tools for handling database versioning in your development platform, or perhaps for your specific DB.
I don't know any of these, but check out this list. I see a lot of pay things out there, but there must be something free. Also check this out.