Tying into mopar's idea of using it as a model for pay I always like to use attorney's as examples. If an attorney took your case you would pay him hourly. Then plug in how many hours they had to work on your case. It's also easy to form an equation using this example as well.
A company makes widgets and sells them direct to the consumer. As it turns out, the widgets are popular, and demand is high. The company raises the price of the widgets since demand is so high (they want to make the most money, right?). These widgets are amazing! Demand is still high. The company raises the price again and people continue to buy the widgets at the even higher price.
At some point, widgets will reach a threshold where consumers will refuse to pay any more. Demand will go down. If the company continues to increase the price, demand will continue to go down. If you're picturing this correctly, you're seeing an downward opening parabola, who's vertex represents the maximum profit (y-coordinate) and the corresponding price the company can charge to generate the highest profit (x-coordinate).
A parabola, of course, is the graph of a quadratic function, which is not one-to-one.
__________________ The light at the end of the tunnel has been turned off due to budget cuts.
It's really two classifications: 1:1 and functions.
Functions have only one y for each x value. So, for example, each of the kids in my family has only one mom and only one dad. (We're not counting, for the sake of this example, Brian's birth parents.) The vertical line test is the standard way a kid can tell from a graph whether or not it's a function. Though, to tell you the truth, I think it oversimplifies things; the kids aren't remembering WHY it works, just testing for that vertical line.
Functions which are 1:1 add the requirement that each y also has only one x value. Kind of like one husband for each wife, and one wife for each husband.
Think of the ads for dating services: there are some that match you up with the one person they think will be perfect (1:1) and some that give you a list of possibliities (from your point of view, a function. Unless, of course, you're also one of several names they also give to someone else.)
As far as actually using them "in the real world"-- I would say that most people don't really use them. The definitions are ways we classify equations and relations in order to make studying them easier.