I'm fairly new to electronics, and I'm trying to understand how and electrical circuit works in an Arduino context. I've set the multimeter to 2K (as the resistor is 330Ω). When the test leads contact the resirtor legs from positive to negative it measures about 430Ω, and 230Ω in the other direction. Why is that? Am I doing something wrong?
You are connecting to a circuit which is powered. Some of that power is going into your multimeter and screwing up the reading.
The multimeter is really a sensitive current meter. The resistance mode applies a fixed voltage to the resistor and measures the current. If there is power on, then that changes the current.
You may also get different values when unpowered. Some of the current supplied by the meter goes into the other components that are connected.
A DMM generates a constant current for each range (up to a maximum of about 2.5 V) and measures the voltage across the terminals. V = I x R so the voltage reading is directly proportional to the resistance. By measuring on the 200 mV range, it avoids biasing most semiconductor junctions.
Unlike the old-fashioned analog resistance meter which had a hyperbolic scale.
I don't know that much about the multimeter itself, but i tried MorganS's advice on disconnecting the power and now the multimeter displays the correct resistance (as shown in the attached image). But what does actually change when I connect a resistor to a circuit? Voltage, ampere, watt or all three of them?