The first thing you should do with a second-hand computer is to erase the internal drive and install a clean copy of OS X. How you do that depends on the model, and on whether you already own another Mac. If you're not sure of the model, enter the serial number on this page. Then find the model on this page to see what OS version was originally installed.
1. You don't own another Mac.
If the machine shipped with OS X 10.4 or 10.5, you need a boxed and shrink-wrapped retail Snow Leopard (OS X 10.6) 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 has less than 1 GB of memory, you'll need to add more in order to install 10.6. I suggest you install as much memory as it can take, according to the technical specifications.
If the machine shipped with OS X 10.6, you need the installation media that came with it: gray installation discs, or a USB flash drive for some MacBook Air models. If you don't have the media, order replacements from Apple. A retail disc, or the gray discs from another model, will not work.
To boot from an optical disc or a flash drive, 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, you don't need media. 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.
2. You do own another Mac.
If you already own another Mac that was upgraded in the App Store to the version of OS X that you want to install, and if the new Mac is compatible with it, then you can install it. Use Recovery Disk Assistant to create a bootable USB device and boot the new Mac from it by holding down the C key at the startup chime. You will need the Apple ID and password that you used to upgrade. Note that if your other Mac was never upgraded in the App Store, you can't use this method.
Once booted from the disc, the USB device, or 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, get 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.
If the previous owner "accepted" the bundled iLife applications (iPhoto, iMovie, and Garage Band) in the App Store so that he or she could update them, then they're linked to that Apple ID and you won't be able to download them without buying them. Reportedly, Apple customer service has sometimes issued redemption codes for these apps to second owners who asked.
If the previous owner didn't deauthorize the computer in the iTunes Store under his Apple ID, you wont be able to authorize it under your ID. In that case, contact iTunes Support.
