I get that the MembershipProvider is in the System.Web.Security namespace. I have added a reference System.Web in my project, I've added a using System.Web.Security directive in my .cs file - but as you can see, VS2010 does not believe me:
The type or namespace name 'MembershipProvider' could not be found
(are you missing a using directive or an assembly reference?)

What am I missing here?
You need to add a reference to System.Web.ApplicationServices.dll.
This is mentioned in the MembershipProvider help:
Namespace: System.Web.Security
Assembly: System.Web.ApplicationServices (in System.Web.ApplicationServices.dll)
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