That Seagate 1 TB unit is SATA II, so if you have SATA ports on your motherboard, you most certainly can install one of these inside your case as a second drive. External units are ideal as backup systems, but you say you are not looking for that - you just want more drive space.
You have a VERY common need and there is a very common process for that. It is called Cloning. The specific need is to upgrade your machine to a much larger hard drive in such a way that the new unit receives ALL of your OS, applications, data, - everything! - and completely takes over the functions of your C: boot drive. This leaves you with a "spare" drive - your original unit. Manufacturers of drives love to sell you a new large unit and help you for free to make the transition. The unit pshoah linked to is by Seagate, so if you buy one of theirs go to the Seagate website and download their free utility called Disk Wizard. Install it on your old drive, and install the new drive in your machine. Then you run the Disk Wizard package to make a Clone. You have to set some choices as you do that. For example, you should choose NOT to use the default setting that simply makes a Partition on the new drive the same size as your old drive - instead, set it to use ALL of the larger drive's space in one volume or "drive". Read the instructions on Disk Wizard to see what you will need to adjust. What it will do is to copy to the new drive (adjusting size etc. as you choose) absolutely EVERYTHING from your old drive and make the new one bootable. Then you shut down and I recommend you disconnect the old drive - both data and power cables - and reconnect the new drive in the same place the old one was plugged in (assuming they both are SATA units). When you boot up your machine will look and run the same, except that your C: drive will have a LOT more Free Space. LATER, after you are sure it is all working as you want, you can reconnect the old drive and use Disk Wizard or Windows' own Disk Management tools to Delete its old Partition and Create a new one that is not bootable, then format it so that the old drive becomes a useful second data storage device.
If you buy a unit from WD instead, their free tools package is called Acronis True Image WD Edition. In fact, both this and the Seagate tool are customized versions of Acronis True Image. Other hard drive makers have similar tools on their websites, but check on that and be sure you can get the free utility you need for the hard drive unit you buy.
Oh, one thing to check on before you start. Any hard drive over 137 GB requires a feature called "48-bit LBA" to be used. That is built into ALL SATA drives and controllers. On the Windows OS side, Win XP did NOT have it in the very first version released, but it was added with its Service Pack 1 and is included in all versions thereafter. So, IF you have only original Win XP with NO Service Packs installed, or an earlier OS, you MUST update your OS currently on your old hard drive BEFORE you go through the installation and cloning onto a new large drive. But if you have at least SP1 installed on XP, or have Vista or Win 7, this is NOT a problem, so ignore it.