Visual Studio is an editor that helps you write .NET programs. If you are a "smart guy", you can very well write .NET applications without using Visual Studio. There are several other free editors exists to help you develop .NET applications, even though most of them are not even close to to the features Visual Studio gives you. If you knwo how to call compiler commands from command promt, you can even use NOTEPAD to write .NET applications.
.NET Framework is far different from Visual Studio. .NET Framework is the "Engine" on which the .NET applications run. A .NET application cannot run on any machine unless it has .NET Framework installed.
If you are a developer, you will probably install Visual Studio on your computer so that you can easily write code and compile. When you install Visual Studio, it automatically installs .NET Framework on your computer so that you can compile and execute your applications.
When you deploy your application on a client's computer, you will need to install only the .NET Framework on the computer. All .NET applications can execute on a computer without having the Visual Studio installed on that machine.