How do I get started designing and implementing a script interface for my .NET application?
There is VSTA (the .NET equivalent of VBA for COM), but as far as I understan
[EDIT: As covered at length in the comments in this, assuming you have a significant need to enable internal scripting where you are hosting snippets or functions someone gives you to customise your app, as opposed to a purely external scenario where one is providing a facaade to allow people to dig stuff out of your app on a more rigid predefined basis]
IronRuby and IronPython are very neat and appropriate for this (but as the other answer says, PowerShell may be appropriate if you have a more infrastructure-type thing).
EDIT: Other ideas for enabling internal scripting are
EDIT 2 June 2011: IronJS may also be a suitable candidate, there's a Hanselminutes that talks it thru.