Macbook Pro (Mid 2009) will not turn on without power cord [new battery installed]
There could be a couple of things going on.

The first, and easiest, solution is to reset the SMC. Apple has a KB article on that:

http://support.apple.com/kb/HT3964?viewlocale=en_US&amp;locale=en_US

Failing that I would then look at the battery as the cause. Remove it and replace the old one. If it then boots (assuming the old one has ANY charge on it) call the seller and get another battery.

Am I missing anything here?