Yes, you can.
You have to implement a shell extension which has to hook the file copy operation. There are plenty of examples of how to do these things in C++ (using the native Win32 API) and also using C# or other .NET framework languages. I'm sure you can use Delphi too but I don't know the details involved in making the Win32 API calls..
Here's an article which details the implementation of a simple shell extension in C#.
Of course, you may also want to take a look at windows file I/O filters, although they only allow you to observe file operations such as creating, copying, changing or deleting. In .NET creating a file I/O filter is trivial using the System.IO.FileSystemWatcher class and if all you want is to detect and log such file copy operations it may be all you need to do.
NOTE If you decide to implement a shell extension in C#, you should know that it will only work on operating systems where the .NET framework has been installed (the version that you used to create the extension) - although most people now have at least .NET 1.1, and 2.0..