The first thing you should do after acquiring a used computer is to erase the internal drive and install a clean copy of OS X. How you do that depends on the model. Look it up on this page to see what version was originally installed.
If the machine shipped with OS X 10.4 or 10.5, you need a boxed and shrink-wrapped retail Snow Leopard installation disc, which you can get from the Apple Store or a reputable reseller — not from eBay or anything of the kind.
If the machine shipped with OS X 10.6, you need the gray installation discs that came with it. If you don't have the discs, order replacements from Apple. A retail disc, or the gray discs from another model, will not work.
To boot from an optical disc, insert it, then reboot and hold down the C key at the startup chime. Release the key when you see the gray Apple logo on the screen.
If the machine shipped with OS X 10.7 or later, it should boot into Internet Recovery mode when you hold down the key combination option-command-R at the startup chime. Release the keys when you see a spinning globe.
Once booted from the disc or in Internet Recovery, launch Disk Utility and select the icon of the internal drive — not any of the volume icons nested beneath it. In the Partition tab, select the default options: a GUID partition table with one data volume in Mac OS Extended (Journaled) format. This operation will permanently remove all existing data on the drive, which is what you should do.
After partitioning, quit Disk Utility and run the OS X Installer. When the installation is done, the system will automatically reboot into the Setup Assistant, which will prompt you to transfer the data from another Mac, its backups, or from a Windows computer. If you have any data to transfer, this is usually the best time to do it.
You should then run Software Update and install all available system updates from Apple. If you want to upgrade to a major version of OS X newer than 10.6, buy it from the Mac App Store. Note that you can't keep an upgraded version that was installed by the previous owner. He or she can't legally transfer it to you, and without the Apple ID you won't be able to update it in Software Update or reinstall, if that becomes necessary. The same goes for any App Store products that the previous owner installed — you have to repurchase them.