To measure the current running through your LED circuit, you must pass the current through your multimeter. The only way to do this is to interrupt the circuit between two components and insert your multimeter, as if it's a circuit component, to complete the circuit.
Switch the multimeter selector to measure DC current in milliamps (mA). Then break the connection between the resistor and LED. (If you are using alligator clips, simply remove the clip that connects the resistor and the LED. If you are using a breadboard, remove the jumper wire.) The LED should turn off.
Next, touch the positive (red) multimeter lead to the unconnected resistor lead and the negative (black) multimeter lead to the unconnected LED lead, as shown. The LED should turn on, because the multimeter has completed the circuit, allowing current to run through it. The current reading shown is 2.14 mA.
Now, insert your multimeter at another connection point in the circuit (for instance, between the positive battery lead and the resistor), taking care to open the circuit at the point of measurement and to orient the multimeter leads with the positive lead at a more positive voltage point than the negative lead. Do you get the same current reading as before? You should, because this simple circuit provides only one path for current to flow.