If there's an operating system installed on the hard drive and everything is functional, then yes, you can start up the Mac without a CD. Hold the 'option' key down as you boot, and you will be presented with icons of bootable volumes.
If you don't see any bootable volumes, then it's likely there's no OS on the hard drive (or another problem, like a disconnected hard drive, or a damaged OS install, etc).
Whether or not there is an OS on the hard drive, though, a Mac computer should never be sold or purchased without the original software restore CDs/DVDs that came with the computer. They're only good for that, specific model of computer, so keeping the disks while selling the Mac is pointless -- in other words, those CDs/DVDs that came with the computer should follow the computer from buyer to buyer, never leaving the computer's side. I would contact the seller of the computer and kindly ask them to complete the sale by including the original software restore CDs/DVDs.
Either way, you will eventually need OS CDs/DVDs. A computer without the original software restore CDs is like a car without keys: useless, for the most part.
