What does it mean when we say an A/D converter has 1 LSB accuracy?
It can't be any more accurate than 1 LSB per sample but it can be a lot worse depending on: -


Intergral non-linearity errors
Dynamic non-linearity errors
Gain error
Zero error
Reference error
Noise


And all manner of drifts with temperature.

All monotonic ADCs have a resolution of 1 LSB but none are accurate to 1 LSB across their whole input range if you take all the things that can go wrong (items in list above).

Several consecutive samples averaged (i.e. over-sampling) can prove an ADC to be better than 1 LSB accurate of course.

So, if you have an ADC that takes one sample it might resolve the sample into a binary number that represents (say) 0 to 2.55 volts. If it is an 8 bit ADC then 1 LSB represents 10 mV and if the input voltage were zero volts then you'd expect an output of 00000000. However if the input were 4.9 mV the output would still be 00000000. As soon as the ADC input rose above 5 mV you would expect the output to be 00000001 and it would stay at that value until the input rose above 15 mV.

In other words the accuracy of the binary number has a range of 10 mV or 1 LSB.