Snow leopard install

Apple no longer sells the retail version of Snow Leopard. If your disk has a gray label, then it is a machine specific version of Snow Leopard which shipped from the factory and it may be very hard to get. You would have to find the appropriate disc online probably on eBay so be careful.

When you look at the bottom of the DVD is it nice and clear or is it dirty or scuffed up? If it has scratches in concentric circles, then that is the worst possible thing to go wrong as the scratches that follow the tracks make it impossible for the disc to be read. If the bottom of the DVD looks good, then most likely the optical drive is at fault. You may be able to use a special DVD cleaning disc which contains a few bristles to clean the lens/laser, but that only works if it has a light coating of dust on the lens/laser assembly. Many times the laser weakens and doesn't have enough power to be able to read the data on the DVD. You can try using a USB DVD drive.

What is the exact model of your Mac? You can get this information by entering your serial number here:

https://checkcoverage.apple.com/

If your Mac is able to run later versions of macOS, then you may have access to Internet Recovery Mode (Command + Option + R) to access the online macOS installer. Depending on the exact model of your Mac it may also be possible to create a bootable macOS USB installer:

https://support.apple.com/en-us/HT201372

Here is an article which can help you determine which versions of macOS are compatible with your specific Mac:

https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

It means either the DVD is dirty or damaged, or perhaps the optical drive is dirty, worn out, or broken.