In your photo, it appears that you have the solar cell leads both plugged into the same terminal and therefore shorted out and doing nothing. It also appears as if you have no current limiting resistor in series with the LED, so I will assume that you are using one of those LEDs with the resistor built-in. This circuit might work if you are using a FET and you apply the solar cell voltage with the right polarity. Can you provide any more info?
Also, I don't recognize that solar panel but I hope it has more than one cell in it, because one cell only gives about 0.5 volts which isn't enough to bias off the FET.