How do I get into dos if a computer doesn't have "Run"?

If in WinXP...
Start -> Programs -> Accessories -> Command Prompt

If you're in 98 or so, open:
My Computer -> C: -> Windows -> MSDos

Should work... Perhaps you should start listing the operating systems that you're working with...

What is dos if not a program?

search your computer for command.com or cmd.exe. execute one of them and you're in dos.