C# -Four Patterns in Asynchronous execution

What you have there is the Polling pattern. In this pattern you continually ask "Are we there yet?" The while loop is doing the blocking. The Thread.Sleep prevents the process from eating up CPU cycles.

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Wait for Completion is the "I'll call you" approach.

IAsyncResult ar = data.BeginInvoke(null, null);
//wait until processing is done with WaitOne
//you can do other actions before this if needed
ar.AsyncWaitHandle.WaitOne(); 
Console.WriteLine("..Climbing is completed...");

So as soon as WaitOne is called you are blocking until climbing is complete. You can perform other tasks before blocking.

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With Completion Notification you are saying "You call me, I won't call you."

IAsyncResult ar = data.BeginInvoke(Callback, null);

//Automatically gets called after climbing is complete because we specified this
//in the call to BeginInvoke
public static void Callback(IAsyncResult result) {
    Console.WriteLine("..Climbing is completed...");
}

There is no blocking here because Callback is going to be notified.

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And fire and forget would be

data.BeginInvoke(null, null);
//don't care about result

There is also no blocking here because you don't care when climbing is finished. As the name suggests, you forget about it. You are saying "Don't call me, I won't call you, but still, don't call me."

while (!ar.IsCompleted)
{
    Console.WriteLine("...Climbing yet to be completed.....");
    Thread.Sleep(200);
}

That's classic polling. - Check, sleep, check again,