Someone else is having the same problem as you over here:
http://www.techspot.com/vb/topic45998.html

Emachines noturiously have bad power supplies - however it sounds like a MoBo fault in this case.
Not sure if you have an ATX MoBo but if you do they usually have a little LED on them that tells you if it is receiving power (there should always be 5v circulating on an ATX MoBo when power is connected to it.)
You could have a look for this - but like has been said - if you have changed to power supply it will either be the power switch or the MoBo - my money is on the MoBo - however have a quick check towards the bottom right corner of the MoBo to double check you have not pulled the power connector for the switch when you replaced the Power Supply.

Good Luck!