Http Context is handled a little differently in C# 3.5 onwards, its a little hard to get the full picture by looking at the msdn docs ("whats with these 2 new context classes???" is what i thought) so here's my take on it.

HttpContext

This is the vintage asp.net context. The problem with this is that it has no base class and isn't virtual, and hence is unusable for testing (cannot mock it). It's recommended to not pass it around as function arguments, instead pass around variables of type HttpContextBase.

HttpContextBase

This is the (new to c# 3.5) replacement to HttpContext. Since it is abstract, it is now mockable. The idea is that your functions that expect to be passed a context should expect to receive one of these. It is concretely implemented by HttpContextWrapper

HttpContextWrapper

Also new in C# 3.5 - this is the concrete implementation of HttpContextBase. To create one of these in a normal webpage, use new HttpContextWrapper(HttpContext.Current).

The idea is that to make your code unit-testable, you declare all your variables and function parameters to be of type HttpContextBase, and use an IOC framework eg Castle Windsor to get it injected. In normal code, castle is to inject the equivalent of 'new HttpContextWrapper(HttpContext.Current)', whereas in test code you're to be given a mock of HttpContextBase.