I want to buy a single POE source and distribute it with cheap network hubs that do not have POE. Will this work?
No. POE will only be supplied to devices that are directly connected to the POE switch, and only by request.
The standard by which this works is described here: http://en.wikipedia.org/wiki/Power_over_Ethernet#Standard_implementation
That won't work -- PoE doesn't carry through additional switches/hubs -- the last switch before the device has to be the PoE switch. You can safely plug non-PoE devices (such as other switches) into PoE ports, though.