To understand cryptocurrencies, you have to first understand what normal currencies are.
Before currencies existed, you had individuals trading and bartering their services with other individuals. Let's say you have Al, Bob, Crystal, and Dave living together in a society. Al is a dentist, Bob is a woodworker, Crystal owns the land rich in timber, and Dave is a blacksmith.
Things are somewhat simple when, for example, Bob needs wood and Crystal wants boxes. Bob can ask for wood from Crystal, and Crystal can provide that wood + additional wood so that Bob can make her boxes from it. But things get screwy when you require too many parties to provide to many separate services for a single product, or where parties do not want what the other parties offer. For example, if Al wants a sword, that sword will require wood from Crystal, and labor from Bob and Dave. If they all want their teeth worked on, then, again, that's simple -- but if just a single one of them decide they do not want Al's services, you have a problem: how to get that person on board so that everyone can benefit?
Currencies are a sort of "economic buffer." Currency allows people to convert their efforts into something that maintains its value and can be converted back into goods or other services at a later point in time. Say, for example, Al wants a chair from Bob, but Bob does't want his teeth looked at. Bob makes the chair, delivers it, and, in return, Al gives Bob a piece of paper that says, "This paper is good for 1 dental exam." That piece of paper now has a value associated with it, because that paper represents a dental exam from Al.
Bob knows that Crystal really needs her teeth looked at, but Crystal hasn't been able to get Al to look at them because she has nothing Al wants. But Crystal has lots of timber, which Bob, the woodworker, wants. In fact, Bob values Crystal's timber more than he values a dental exam. So Bob agrees to give Crystal his piece of paper from Al in exchange for timber.
What just happened here?
Al got the chair he wanted from Bob, and Bob converted his efforts into an "IOU" from Al, which he then traded to Crystal for timber, who then received a dental exam from Al. In fact, if Al's work becomes in enough demand, then slips of paper that entitle the holder to dental exams become very valuable.
This is where the idea of "counterfeiting" comes into play. Counterfeiting is harmful because when people begin to convert their time and effort into IOU's, then some people make fake those IOU's. Why? Let's say it took Bob 10 hours worth of effort in order to earn 1 IOU from Al. That IOU then has a value equal to 10 hours of Bob's work. So if Dave wants a dental exam from Al, but has nothing to trade for an IOU, him counterfeiting an IOU unfairly gives Dave the ability to falsely convert his efforts into something valuable. After all, if no one stops Dave from doing what he's doing, why would anyone actually earn a real IOU from Al (which requires actual effort)?
For this reason, currencies have largely relied on stuff from the real world that is scarce or finite. What does that mean? Take gold, for example. Gold is rare and, at this point of time, only naturally occurring in large quantities. This means if someone is holding gold in their hand, there are only 2 possible ways that could have happened: (1) they got it from nature, (2) they got it from someone else who, at some point, got it from nature. Another valuable part of gold is it is highly divisible... meaning, based on weight, you can take a gold nugget and very precisely value things based on different weights of gold. IOU's don't have this... you can't really cut an IOU in half and expect Al to give half an exam to one person and half an exam to another.
So now, when Bob makes a chair for Al, Al has the ability to value that chair based on a quantity of gold. When Bob get's that gold, he can then use it to trade with other people who also value gold. Whether Bob found the gold or earned it, gold is gold, and the end-receiver of it can easily test to see whether it is real gold, and thus test to see if it is a real representation of value (instead of counterfeit value).
But gold has its problems. It's heavy. It's easy to lose. And it can only scale so high due to there only being so much of it in the world.
For this reasons, countries began to centralize their gold reserves and, instead, issue currencies that were based on gold, but not gold themselves. They were, essentially, IOU's again. But not just any IOU... IOU's that were offered by sovereign powers with lots of respect amongst the world... these powers put in the effort to make it hard to counterfeit those IOU's, and even dedicated large parts of their authority to stop people from counterfeiting (sometimes by simply killing anyone who attempted to do so, because it was that dangerous to the reliability and trust that people had that a piece of paper represented real value).
Fast forwarded a bit more, and we have largely gone off the gold standard. This means that, in the US for example, currency no longer is an IOU for actual gold. Rather, currency is, itself, the resource. The government "mints" currency, which means the government creates scarcity in the currency, as well as protects against people counterfeiting it. This trust and value gives the currency its value.
So if you hold up a US $20 bill, that $20, itself, has a value of $20 to anyone who has trust in the United States government. If you are paid $20 per hour, that bill represents an hour of your time converted into a piece of paper that is very hard to duplicate. In fact, if you were to work for an hour, get paid $20, and then burn that $20, you have created an unbalance in the system because you put into $20 of value, converted it into paper, and then destroyed the paper containing the value. Likewise, if you find $20 on the street, that money represents value that someone else earned and lost, which is a "wind fall" for you who now has gained $20 in value without any effort.
WITH THAT SAID...
Cryptocurrencies are a digital form of currency, which is a big deal. Remember, throughout this discussion, the biggest issue has been "counterfeiting," and when it comes to digital anything, it is very simple to make a copy of things. In fact, even if you don't mean to, you will create copies of things (copying a file from your computer to your hard drive creates a duplicate). This ease of duplicating information has made it hard to make a currency, because the second the currency becomes popular, it will become easier to fake it than earn it.
However, cryptocurrencies have created a way for people to convert their efforts into a digital token system, and those tokens have proven to be secure from duplication. In other words, they are the digital equivalent of gold. In fact, because there are also a limited amount of tokens, they, like gold, provide that important part of a currency where you need to know "how did someone get this?"
This means that if you own, say, a bitcoin... you either (1) mined it yourself, or (2) got it from someone who, at some point, mined it. This makes cryptocurrency very trustworthy as a means to "hold value" that it converts. Additionally, because it is "decentralized," no single group is allowed to "print more of it," or be "taken down" (which would make it useless). Remember Al? If he goes out of business, then his IOU's are worthless... you can't just use them at another dentist. This sense of reliability/dependability is why only the most powerful, stable countries have secure currencies others are willing to trust their value with.
The difference between the various cryptocurrencies comes down to small changes in how they operate, to try to make some more attractive than others, but at their core, they all share the ability to act as legitimate currencies.
"Mining" is just like mining for gold. It is the only way to introduce new "stuff" into the system, which can then be treated as currency. In the case of bitcoin specifically, mining also helps keep the whole system working and secured.
tl;dr: Cryptocurrencies are digital equivalents of currencies, and mimic natural limitations via very sophisticated programming.
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