If you want to keep an old computer and don't have extras, you will need a keyboard, mouse and monitor as well. Also, if you don't have cables, make sure that either your motherboard or HDD come with cables (SATA or PATA/IDE depending on the kind of drive)
Full list:
CPU
Motherboard
RAM
Power Supply
Hard Drive
OS
Disk Drive (Recommend at least a DVD reader - but burners are so cheap it's hard to resist)
Graphics Card (for gaming, or if your motherboard does not come with integrated graphics)
Case
Peripherals:
Keyboard
Mouse
Monitor
Optional Components:
Second graphics card (Crossfire or SLI)
Speakers and/or headphones
Additional Hard Drives
Additional RAM
Additional Disk Drives
Multiple monitors
All you need for the core of your computer is in the first list.
necessary peripherals are in the second.
things that would be nice on a high budget are detailed in the third.