You basically have three options for connecting your computer to a broadband connection:
1. Internally - This tends to be the neatest solution if you only have one computer to connect to a broadband connection, and is a good choice for ADSL. I'm not aware of any internal "cable" modems. To use an internal modem, you'll need a spare PCI slot inside your computer (they're white, and most modern PCs have about 4-6)
2. USB modem - This option is favoured by most broadband suppliers as it is the easiest to set up. However, it is not as neat as an internal modem, and uses more system resources, which is not good for online gaming. All you need for this solution is a spare USB slot.
3. Ethernet - This option can come in the form of a modem with a network connection, or a router, which allows multiple PCs to be connected to one broadband connection. This type of connection is the quickest in theory, and requires no dialling up. All you need for this type of connection is a spare network port on your PC.
Seeing as you bought your PC in 2001, I would imagine that you have all three of these options available to you. As long as you can get broadband, I wouldn't have thought your PC would be holding you back.
These are the MINIMUM requirments off the BT site PC
Computer hardware Your computer must have all of the following: Pentium II 233 (or equivalent) or above 64 MB RAM CD-ROM drive USB port 40 MB available hard disk
A compatible Operating system Windows 98/98SE Windows 2000 Windows XP Windows Me
Note: The minimum operating system is determined by the broadband modem you purchase. Generally, it must be Windows 98 or above. The above operating systems are supported by the broadband modem available from BT. Windows NT and Windows 95 are not generally supported.
Computer hardware Your computer must have all of the following: PowerPC G3 300MHz (CPU) 64MB (Memory) 40 Mbytes available on hard disk or more USB port CD-ROM drive
A compatible Operating System Mac OS 8.6 Mac OS 9.x Mac OS X (10.0; 10.1; 10.52x