The Schmitt trigger works similar to a comparator, so imagine you have a comparator with a Vref in the non inverting input. When the input voltage falls below that threshold, the output of the comparator saturates high. If the input voltage then goes above the threshold, the output goes low. If you were to put the Vref into the non inverting input, the same rule applies, but backwards, so when the input goes below the threshold, the output goes low, and if it then goes above, the output goes high.
The general rule is if the non inverting input is less than the inverting input, the output will be high.
The Schmitt trigger works in almost the same way, except it has upper and lower limits. As you can see from your diagram, the input voltage (sine wave) starts at 0, which means the inverting input is lower than the non inverting, hence the output starts high. Once it goes above the high threshold, the output goes low. With a Schmitt trigger, the output will stay low untill the lower threshold is met.
As with the comparator, if you were to swap the input voltage to the non inverting input, it would be the other way round.
I believe this is what you were asking, but please correct me if I am wrong as the English was rather poor, as Olin pointed out.