I'm trying to wrap my head around the 20m and the 2000µ setting. On 20m I see 0.01 but on the 2000µ setting I see 005. Does this mean my circuit is using 5µA of current? Why the discrepancy?
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This multimeter looks suspiciously similar, look at its spec sheet. It is important to realize that every meter has an accuracy and it varies per range. At 2000µA the meter's specification is +/-1.0% +/-5 counts. (Check the manual that came with your meter for its own specifications).
This means your reading 005 can vary:
The same accuracy applies for the 20mA range. This means your reading 0.01 can vary:
This clearly shows that you always want to use the range that is as close as possible to the measured value. It also shows that the last digit can often only be used to view the trend of a measurement rather than measuring its absolute value.
This specification does not account for the burden voltage as mentioned in other answers. The internal resistance of your meter influences the circuit and with that may influence the measurement too. Unfortunately the higher the resolution of the range, the higer the burden voltage becomes. So if you switch your range to an accurate range, the meter influences the circuit with a relatively large internal resistance.
Depending on the circuit you measure, the internal burden resistance may influence the current you are trying to measure.
Especially on low current settings the burden resistance gets larger and has a bigger influence.
But as others have noted in 20m you are hitting the accuracy limit of your DMM.