In general, a comparator is any circuit whose output will assert when an input signal rises above a given threshold, and de-assert when the input falls back below the threshold. (In our two example plots, the output logic is inverted, so when the output signal asserts it is driven LOW, from a normally HIGH state).
Figure 1 shows a plot of single threshold comparator input/output relationship (input is shown in blue, and output is shown in gold). You can see that the output changes once the input voltage crosses the threshold voltage level. You may have also noticed that the transition edges of the output signal are somewhat noisy. When an input voltage is near the threshold voltage, sometime noise from the input circuit (in the mV range) is enough to push the input above or below the threshold. The noise close to the threshold transitions is from the output signal switching back and forth between an asserted and de-asserted state.
With a two threshold comparator like the Schmitt trigger is used (where the output asserts once an input signal rises above a upper threshold, and does not de-assert until the input signal falls below a second lower threshold). Once the first threshold is reached, any noise on the input signal would have to push past the opposite threshold to change the output. Overall this makes a two threshold comparator more resistant to input noise than a single threshold comparator. Figure 2 shows the input and output of a Schmitt trigger (input in blue, and output in gold as before); you can see that there is no noise on the signal transitions.