As we have generally slow changing DATA signals i.e. DATA signals have slow rising and falling edges. And use of these signals creates problems in the working of Digital circuits. Hence to avoid these problems we use SCHMITT TRIGGER to sharpen up the edges of DATA. The transfer characteristics of this trigger are as follow:

We have only 2 output voltage levels Vo- & Vo+ and there are 2 input voltage thresholds V1 & V2. the SCHMITT TRIGGER follows the above characteristics i.e. when ever input voltage is increased from 0, we have the output voltage as Vo- and output remains the same till input voltage is less than V2. Hence when ever there is an increase in the input voltage, transition would occur only when input voltage becomes greater than V2.
When-ever input voltage is decreased from the high value (>V2), we have the output as Vo+ and the output remains the same until input voltage becomes less than V1.
Hence behavior of SCHMITT TRIGGER depends on whether input is increasing or decreasing and trigger is said to give hysteresis of V2 – V1. This is used to eliminate the effect of noise on the signal. We illustrate this fact using the following example.
Suppose we have the following signal with superimposed noise and we take two cases: with small hysteresis V2 – V1+ and with large hysteresis V2 – V1—and notice the effect of SCHMITT TRIGGER

Small hysteresis V2 – V1+: As voltage is less than V1+, hence initial output is Vo-. When the voltage increases to greater than V2, output jumps suddenly from Vo- to Vo+. Now output voltage remains same till input voltage drops below V1+. When voltage drops below V1+, we get the output as Vo-. After some time input voltage again increases beyond V2 and hence output again jumps to Vo+.
Large hysteresis V2 – V1–: As voltage is less than V1- , hence initial output is Vo-. When the voltage increases to greater than V2, output jumps suddenly from Vo- to Vo+. Now output voltage remains same till input voltage drops below V1+. When voltage drops below V1+, we get the output as Vo-.
We represent the outputs of SCHMITT TRIGGER below:

And we can see that we get a clean square wave for large hysteresis while for small hysteresis we get many transitions in the output but it is still better than the output without using SCHMITT TRIGGER.