There is only one path, so the same current flows through the battery, the switch, the resistor and the LED. Close S1 and the loop is complete; open it and the current stops everywhere at once. That single shared current is what makes it a series circuit.
An LED is a diode. Once the voltage across it reaches its forward voltage, about 1.8 to 2.2 V for a red LED, its current rises very steeply for any further increase in voltage. Connected straight across 9 V it would try to draw far more current than it can survive. R1 takes up the difference: the 9 V splits into roughly 2 V across the LED and 7 V across the resistor.
With 7 V across 330 Ω, Ohm's law gives about 21 mA, close to the continuous rating of a common 5 mm red LED. If you want the LED and the battery to last longer, 470 Ω gives about 15 mA and the LED looks only slightly dimmer.
The order of the parts in the loop does not matter: the resistor can sit before or after the LED and the current is the same. The LED's direction does matter. Current flows from anode to cathode, so the anode (the longer lead) must face the battery's positive terminal, or nothing lights.