I guess you have a Retail DVD for Mac OS X 10.6 Snow Leopard? If so, the computer you want to use it on, has to be released in 2009 and earlier (2008, 2007 and 2006), since that installer of 10.6.3 does not include drivers for 2010 and 2011 and 2012 and 2013 Mac hardware.
If all is aligning, have you formatted the new HDD yet?
Can you be more specific? The A1278 model Macbook covered many years and generations of MacBooks. Do you know the model identifier? Is it a Macbook 5,1? Something like that will help us narrow it down to what specific model you have, and that may relate to whether or not you can use the retail DVD that you have.
That MacBook never came with 10.6.3, of course.
The commercial 10.6.3 DVD should install without any problems.
However, the error that you get may mean that you have a 10.6.3 DVD that shipped with another Mac, such as a 2010 MacBook, or a 2010 iMac. Those came with installer DVDs that were special builds of 10.6.3, different from the commercial install, because of new hardware in those 2010 Macs.
Is there any possibility that the DVD that YOU have, really came with one of those 2010 Macs?
Where did you get that DVD?
A new DVD for 10.6.3 is only $20, direct from Apple. http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard