Xbox 360 development is largely split into two distinct sections:
XNA Development
This involves using the XNA Framework to develop games/applications, and is done through AppHub. You can find the tools for development (XNA Game Studio) here. Anything developed using these tools must be written in C#.NET and can only be released onto the Xbox LIVE Marketplace as an 'Indie Game' (after passing the review process, of course). To debug your XNA app on the console itself, you need a membership. Another thing to note is that Xbox 360 apps can't access the Internet at all - all communication goes through Xbox LIVE or System Link.
Professional Development
This covers the development of complete Xbox 360 titles, and requires that you be part of the Xbox 360 Registered Developer Program. Once a development license is obtained (and the NDA is signed), Microsoft provides developers with access to the official Xbox 360 SDK and development hardware, which includes full Visual Studio integration and documentation for developing native Xbox 360 applications and libraries using C/C++. The development kit hardware comes in a few flavours and supports debugging from Visual Studio via ethernet, DVD emulation via USB, etc. For more information on the professional development program, see this information.
Developing 'normal' applications for the Xbox 360 isn't really possible at the moment due to the limitations of the XNA Framework and community developer program. If you're part of the registered developer program it might be possible, but there's no way to be sure (due to Microsoft's strict NDAs).
As for Silverlight, there have been some rumours regarding Silverlight on the Xbox 360 but Microsoft hasn't confirmed or announced anything.