Hello,
Zero check is a feature that notes the internal voltage offset within the unit.
Zero correct allows you to algebraically subtract the internal voltage offset value from your measurement thereby correcting the value of zero within the instrument. Very similar to a REL function, however REL can be used to subtract out ANY level of signal or reading. Zero check/correct is specifically for correcting the value of zero by subtracting internal voltage offsets that would otherwise result in the instrument reading slightly higher than "true" zero.
Zero Check and Zero correct should both be done prior to making a measurement. Zero check willl be disabled to perform a reading.
See the manual I'm attaching below for more info. Ref pages 2-13 and on.
Model 6514 System Electrometer Instruction Manual | Tektronix