If your PSU does not have enough power you will probably be able to run your system just fine at the desktop.
As soon as you start a game or any GPU intensive task however one of two things are likly to happen.
If your PSU is over volt protected, it will simply shut down.
If you have a bargain basement PSU it will probably fry and could very well damage other components.
The 4870 will require two 6 pin PCIe power connectors to run.
Most modern PSUs will have these connectors and, if not, you can use a MOLEX to PCIe power adaptor.