For a fantastic explanation and lesson on percentage, check out this post at Math is Fun!
In case you didn't check out that site, a percentage is essentially some sub group of, or a collection of other groups including, some other number. For example, check out these flamingos from the Sacramento Zoo:
I count 11 flamingos in this picture; however, they are only a fraction, or percentage, of all of the birds in this picture. Within the flamingo population, a percentage of them are holding their heads up.
Of the 11 flamingos, 5 have their heads up; hence, 5 / 11 is the fraction of flamingos holding their heads up. As a percentage, that becomes 45.45%. In other words, 45.45 percent of the 11 flamingos are holding their heads up. Got it? Easy? Of course! But that's just the beginning of our story.
When it comes to research, this group of flamingos would be seen as a "test selection group" on which the research is based. Is it fair for me to conclude, based on this one photograph, that about 50% of all flamingos will have their heads held up at any given time? Of course not. In fact, here's another picture of those same flamingos with a few more of their flamboyance (that's what you call a flock of flamingos, seriously) showing most of them with their heads down:
The trouble is, a lot of societal research and polling is done in this manner - form a conclusion from data obtained by polling or studying a very small subset of a general population over a specific time frame, often with a bias or outright mission to prove your original hypothesis on the subject matter. I could go on and on about this nonsense, but that's not my point, so let's get back to that - percentages!
In the second photograph, I count 26 flamingos. I might be wrong, and the exact number isn't what's important here. What if these were sales numbers? Your boss might say something like, "in this quarter, we made X% more money" or "we had an X% increase in sales!" But do these statements really make any sense? That depends... does your boss know how to do math?
If this second photo is the entire flamboyance (seriously!) of flamingos, then the first showed a percentage of them: 11 / 26 * 100% = 42.31% to be more precise. We could also represent the increase in the number of flamingos as a percent increase. Here's the process:
Original number of flamingos: 11
Final number of flamingos: 26
Difference: 26 - 11 = 15
Percent increase: 15 / 11 * 100% = 136.36%
So we can say that the first picture showed 42.31% of the entire flamingo flamboyance, while the second showed a 136.31% increase in the number of flamingos photographed. Does that make sense?
One thing to always remember about percentages - they involve multiplication, not addition. You typically say "a percentage of" where "of" implies multiplication. That's why statements involving terms like "more" and "less" are so confusing - they imply addition and subtraction. The choice of words is incredibly important when trying to get your point across to someone; choosing the wrong words can often lead to you sounding like an imbecile.
When someone says "X% more" they really mean "(100 + X)% of." Similarly, "X% less" should be interpreted as "(100 - X)% of." As an example, if I wanted to take a picture of 10% more flamingos in my next photograph, then I need to find a flamboyance (sorry, it's just so fun to say) containing 110% * 26 = 28.6 flamingos. Would that include a teenage flamingo, or maybe one missing a leg? To test that, we can once again calculate the percent increase: (28.6 - 26) / 26 * 100% = 10%.
If you want a more practical use of percentages aside from calling your boss out on his stupidity, take shopping sales: if an item is part of a 70% off clearance, the sale price will be 30% of the original price. When using a 10% off coupon on top of that, just compound the percentages. The new price now becomes 30% * 90* ORIGINAL. Of course, these new numbers are found by subtracting the discount from the maximum of 100%: (100 - 70)% * (100 - 10)% * ORIGINAL. You never add these discounts together, rather, you multiply them.
Another obvious use is to calculate your tip at a restaurant (which you should be doing, even though it is a stupid practice that should be abolished, and servers should be paid a full wage by their boss). Let's say your meal cost $53.67. How do you figure out the tip? It's really easy...
10% = 0.1 * $53.67 = $5.37 (Let's just use $5.40) <- This is a crappy tip. Don't be cheap.
20% = 2 * $5.40 = $10.80 (just double the 10% amount!)
If you think 15% is a good number, then give somewhere in the middle of those... like $8.50. This is really closer to 16%, but you just did that in your head, showing your mathematical superiority to your friends. See, percentages are simple and useful.
The real moral of the story is to always double check your numbers and make sure you are correctly representing data; otherwise, your entire point may be seen as invalid. Sadly, this type of ignorance is not only found in in the non-technical, but can regularly be seen in otherwise great research articles and technical documents.
