Do I need a serial number to reinstall Mac OS X?

Mac OS X doesn’t have (nor had in the past) Serial Number or “Legit Purchase” mechanism. You can buy/borrow/lend/steal/copy a copy of Snow Leopard (or any other edition) and install it on any number of Machines.

Technically this is possible, however, according to the EULA and all the legal stuff, you can’t; in any case, the operating system doesn’t have any copy-protection mechanism.

EDIT: Bear in mind that a plain OS X Snow Leopard disk is not the same as a recovery disk. Although a recovery disk contains the OS, it is usually tied to a particular model, so if you try to recover a Macbook Pro with the iMac recovery disk, that might not work. (This has happened to me in the past trying to recover the other way around, a friend’s iMac’s HDD failed and he replaced it but couldn’t find his recovery disk. We tried to use my Macbook Pro’s disk but it wouldn’t work. The same with my Mac Pro.

Going to an Apple Store will possibly help in that case. In the end I found my Snow Leopard’s original disk and we used that to install his iMac.

There's no serial number to enter when installing Mac OS X.

The only Mac OS X that does have serial numbers is Server versions 10.6 and earlier. They know are licensed for 10 or unlimited users.

The only problem you may run into is that if you are trying to do the reinstall with a system disk rather than a plain old Mac OS disk.

Say that other computer is a iMac and you are using the system disk from that one (one of those gray disks). The installers do have a little bit of code in them that check machine types, and may not let you install. Or, you may be able to install the OS, but none of the other applications (mainly iLife, etc).

This is not to hard to get around, since the main installer is what is checking - if you dig down on the disk, you can find the actual installer, which should work.