I just had this exact same problem. A lot of people have been suggesting to use an external hard drive to put the Snow Leopard installation media on. This can be slow (eww, USB), and not everyone has a large enough external drive.
All I did was re-size the hard drive inside my Macbook to create another, smaller partition (10GB) and put the Snow Leopard installation files on there. Here's how:
Create the partition
- Open Disk Utility
- Select the Hard drive you want to resize and put the installer on. This is not 'Macintosh HD', in my case it is '260.06 GB Hitachi HTS5...'
- Select the 'Partition tab' and decrease the size of your main partition (In my case 'Macintosh HD') by 10GB. You can do this by dragging the partition up, or by manually changing the value in the text box.
Press the + (plus) to create a new partition. The size of this new partition should be 10GB, it should be formatted with 'Mac OS Extended (Journaled)' and the name should be something snazzy like "Snow Leopard Installer". All up, it should look like this:

- Once done, hit 'Apply'. This should take 5-10 minutes, but don't worry if it takes more (especially with a larger hard drive). You should leave your system alone whilst it is applying the changes, and it should not take any longer than 1 hour.
Copy the installation media
Booting into the installer
- Once you have a Snow Leopard Installer partition set up, restart your computer and hold the 'Option' (or alt) key at boot up and select 'Snow Leopard Installer' from the list. The Snow Leopard Installer should successfully boot up and you should be able to upgrade to Snow Leopard!