NSMutableHTTPURLRequest, a category of NSMutableURLRequest, is how you set up an HTTP request. Using that class you will specify a method (GET or POST), headers and a url.
NSURLConnection is how you open the connection. You will pass in a request and delegate, and the delegate will receive data, errors and messages related to the connection as they become available.
NSHTTPCookieStorage is how you manage existing cookies. There are a number of related classes in the NSHTTPCookie family.
With urlopen, you open a connection and read from it. There is no direct equivalent to that unless you use something lower level like CFReadStreamCreateForHTTPRequest. In Objective-C everything is passive, where you are notified when events occur on the stream.