Charging lithium polymer
First of all it says the green led is powered when the battery is fully charged, but I see nothing controlling it, just goes straight to ground.


The green LED (D3) shares a current-limiting resistor with the red LED (D2) and the transistor (Q1). When charging is occurring, about 400 mA is flowing through D1 and/or Q1, which turns on the red LED. Since a red LED has a lower forward voltage drop than a green LED, this effectively keeps the green LED from lighting until the red LED is turned off. This occurs when the charge current through D1/Q1 drops to "almost zero when the cycle completes".


  Second of all I was going through the datasheet and found that when pin 1 is high the chip will charge and when low it won't charge. So in the circuit it's tied straight to 5v, does this mean that it will not stop charging until pin 1 is low?


No, the IC will stop the charging on its own; you don't need to bring pin 1 low. Pin 1 is there in case you wanted to inhibit charging for some reason.


  And lastly it looks like a pretty simple circuit for charging a battery. Will this circuit safely charge the battery?


I don't see why not. I assume this is from the manufacturer's application notes, and they wouldn't want to present their product in anything other than the best possible light.