You would create the dictionary in the code behind - quite frankly whether it's Win Forms or ASP.NET or pick your .NET library type of choice, the library type is immaterial.
It's very useful when you want to store something with a well defined key that you can look up later. A classic example of a dictionary in ASP.NET is the Session object which can store an arbitrary object (known as the Value) using a meaningful Key.
In your example, you'd create the dictionary using
Dim myDict As New Dictionary(Of String, String)
You would add typically add values to it using
If Not myDict.ContainsKey(theKeyToAdd) Then
myDict.Add(theKeyToAdd, theValue)
End If
Finally, you can retrieve the value using
Dim value As String
If Not myDict.TryGetValue(theKeyToRetrieve, value) Then
Console.WriteLine("The key could not be found")
Else
Console.WriteLine(theValue)
End If