Conversion time is the time it takes for ADC to convert analog input to digital output. Sampling rate is how many times the signal is sampled each second.
Adding more to the answer of @Palpurul, Normally the conversion time should be less than the time between the two samples are taken. For example, if your sampling rate is 50 Hz, i.e signal is sampled at every 20 ms, the conversion time should be less than 20 ms. If not, you may loose the next sample as your are still converting the previous sample value.
This depends on the converter architecture. Pipelines
can have sample rate > conversion time, because you
are marching multiple samples down the pipe. A SAR type
could go almost to conversion=sample (plus setup/hold
and maybe S/H settling time).