This sometimes is the symbol that we use for the quantile function.
https://en.wikipedia.org/wiki/Quantile_function
The very short version of what the quantile function is, is that it is the inverse of the CDF. If you're familiar enough with the CDF, then perhaps this explanation is all that you need.
As for how it's used, well, there may be a bunch of different answers depending on exactly what you mean by "use". In my experience, students often have a very different concept of something being "useful" than the concept academics have.
But I think this will count as a use:
If you give the quantile function numbers which are generated randomly from a uniform distribution, the resulting values will be distributed like random draws from the original distribution. That is to say, if we call U the standard uniform distribution, and if we make draws u1, u2, ..., un, then the numbers Q(u1), Q(u2), ..., Q(un) will be distributed approximately like the distribution of X, the random variable which defines Q.
So basically, since a computer can simulate U, and we can hopefully compute Q, then we are able to simulate any distribution of a random variable.
