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(I'm newbie) I have an Arduino Uno connected to a ULN2803(5V) which lights up an array of LEDS. I would like to take one of the outputs from the ULN2803 and use it as input to the GATE of a MOSFET(P30N06LE) to light another LED(or motor etc). I can get the MOSFET to light the LED without the ULN2803 but it does not work when the ULN2803 pin is connected to the MOSFET gate. What am I doing wrong?

V_GS(TH) of the P30N06LE is between 1 and 2 V. At 1.6 V, you can probably light on a red LED with a reasonable 20mA through it, but 1.6V may be not enough to fully open the MOSFET. Note that generally, the V_GS(TH) is defined when the current reaches 1mA, so you need more that this bare minimum. Typically, that MOSFET can let a current up to 20A with 3 V at the gate (figure 7 of RFP30N06LE), so, the Arduino should be able to supply enough voltage without using the ULN (the number in your title differs to the number in your text). Can you measure the V_GS at your MOSFET with a multimeter?

Oh, and to protect it, if you use an inductive load, such as a motor, add a diode in reverse, in parallel to your inductive load, to provide a path to back EMF.

Final note, the "load" should be between the drain of the MOSFET and the + side of the DC power supply, while the source pin of your NMOS should be connected to the - side of the DC power supply. And avoid using a USB power supply, they may be unable to handle back EMF.