When I measure a voltage of a 9V battery connected to a circuit and then disconnected, it varies. According to my electronics teacher, the voltage should be the same, since I am measuring the battery. I attached my circuit. (the V is the voltmeter) Is it because the circuit is so simple?

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I think your instructor is in error. Batteries have a internal resistance that is in series with the battery terminals. This internal resistance varies with the state of charge of the battery and the resistance increases as the battery discharges. So even with the load of 270 ohms which represents a load of approximately 30Ma. The voltage on the battery terminals will vary with the load connected. You can determine the value of the internal resistance by measuring the battery voltage without a load, then with the known load. subtracting the load voltage from the open circuit voltage and dividing by the load current. The result will be the internal resistance of the battery.

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A battery can easily be modeled as a thevenin or norton equivalent circuit which is a constant voltage source in series with an internal impedance (or parallel for norton). This model should agree with your experimental values.