I mean, really, what is a percent?

We're going to keep coming back to the following statements:
 

All that a % character means from a mathematical standpoint is "some goofball multiplied me by 100."

A % character also implies that the quantity multiplied by 100 resulted from a division of some sort between 2 similar measures. What those measures were can be answered by asking the question "THAN WHAT?"
 

Why would someone just up and multiply a poor defenseless number by 100?

I don't have a definitive answer, but I suspect that it is rooted in early childhood trauma surrounding the decimal point, which conjures up scary mathematical concepts like long division, fractions, rounding errors, infinitely repeating decimals, and from there it's only a short trip to transcendental numbers. Most people are comfortable talking about integers and not quite so comfortable talking about fractional portions. Nobody says "I paid one point sixty five dollars," or "I paid one and thirteen twentieths dollars." Most people would phrase it as "I paid a dollar sixty five" or "It was around 2 dollars."

Yeah, but what does percent mean to me?

We hear the word percent a lot, and use it a lot, even in conversation. To some extent, we understand what a percent, is even if all that calculation stuff gets confusing. It's a way to represent how two things are related to each other by representing one of them in terms of the other, and generating a number to represent that relationship. 100% means that whatever you're comparing is basically the same. 0% means that what you're comparing is radically different. 200% also means that what you're comparing is radically different, just in the other direction.

How do we know what something is a percent of? The short answer is that we don't, not unless someone tells us, or it's somehow implied in what is being represented. If you have a radio or TV, then you've certainly heard the following sorts of claims: "30% less noise", "50% faster" and "10% less fat." The first question should always be "THAN WHAT?"  Asking "THAN WHAT?" isn't just practical. It can be fun, too.
 

CLAIM THAN WHAT?
30 % less noise A jet engine test facility?
50 % faster A slug with a slow metabolism on qualludes?
10 % less fat A 50 gallon drum of lard?
100 % more vitamins A bucket of sand?
20 % more power A can of Spam with electrodes sticking out of it?
tastes 10 % better Lead based paint?

The other way most Americans are familiar with a percentage is through shopping. The United States is not the leading consumer of goods and services on the planet without most of its members having run into sales tax, and the signs "BUY TWO, GET ONE FREE", "$5.00 off", "TWO FOR ONE", "25% off" or "TRIPLE COUPONS" at some point. At some level, most people realize that 5% off a 50 cent candy bar isn't much of a deal, but 5% off of a $200,000 house is a lot of money.

Multiply by 100? Divide by 100? How do I keep them straight??

If there's a trick to dealing with percent math problems, it's knowing when to divide by 100 and when to multiply by 100, and when to add or subtract what you're comparing against from the result. A good rule of thumb is if you see a percent sign, you're going to divide by 100 to get rid of it. If you want to put a percent sign on something, you're going to multiply by 100. And remember, the only thing "%" really means is that the number in front of the percent was multiplied by 100.

At any rate, here are a few more examples of how to use percents. These go into more detail about what percentage actually means than the examples on the previous page.