Do You Really Want To Do That...?
There are several options, but first let's talk about NOT copying everything from the old hard drive to the new one. When you use a computer for a couple of years, it tends to build up what I call Cyber Sludge, and you really don't want that nasty old sludge on your shiny new hard drive.
In most cases, all you really need to copy are the personal files you have created, such as word processor documents, spreadsheets, photos, etc. You can use File/Export to dump your web browser favorites into a plain-text file, and the same applies to your emails and address book.
Various Ways to Transfer Files
To transfer to the new computer, you can burn a CD containing those files, or copy them to one of those nifty USB-connected portable hard drives (also called Flash Drives or Memory Sticks), or attach the files to an email and send them to yourself.
If you have a home network, it's even easier. Just make the hard drive in the old computer a shared drive, and you'll be able to copy the files to the new drive with drag and drop. Without a network, you can install the old hard drive in your new computer, and just access the files on it as your D: drive. If you're timid about popping the hood on your PC, you can take both to a computer repair shop and ask them to install the old drive in the new machine.
Copying the Entire Drive
Since that's what you asked about, yes... there are ways to copy your old hard drive, lock, stock and bearings, to a new hard drive. In the old days, you could use the XCOPY command from a DOS prompt to copy everything in one swell foop, but modern operating systems make this impossible. I recommend you use software specifically written to deal with all the intricacies of dealing with partitions, copying the Windows registry, and handling locked, hidden & system files. Two programs I recommend for this purpose are VCOM Copy Commander and Partition Magic.