Quota sampling is very similar to quantitative stratified sampling; the difference lies in how the participants are chosen once the stratification takes place.
For example, imagine we wanted to know how our presidential candidate was faring with males and females in the United States. The first thing we would need to know is, percentage- wise, the distribution of gender in our country and then ensure our sample has the same distribution. In this case, I happen to know that approximately 51% of our citizens are female; given that, I would want a sample with the same characteristics, 51% female and 49% male. Unlike stratified sampling, where I would randomly select potential voters from the population, in this case I would identify a population of potential voters and recruit participants until I had the sample size I wanted with the proportions called for.