What are Currencies?
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 Greg, Bob, Laura, and Eggert living together in a society. Greg is a Dentist, Bob is a Carpenter, Laura owns land rich in timber, and Eggert is the town's blacksmith.
Things are somewhat simple when, for example, Bob needs wood and Laura wants a cutting board. Bob can ask for wood from Laura, who than provides the necessary wood for Bob to make her cutting board. Transactions become difficult when a single product requires multiple services or items from multiple individuals. Another factor is when an individual doesn't want the product or service offered (i.e. customer prefers Coca-Cola vs. Pepsi).
Greg wants a sword. That sword will require wood from Laura, and labor from Bob and Eggert. If they all want their teeth worked on, then, again, that's simple -- but if one individual didn't want Greg's services, you have a problem. How to get that person on board so that everyone can benefit?
Currencies are a sort of an "economic buffer." Currencies allow 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, Greg wants a chair from Bob, but Bob does't want his teeth looked at. Bob makes the chair, delivers it, and, in return, Greg 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 Greg.
Bob knows that Laura really needs her teeth looked at, but Laura hasn't been able to get Greg to look at them because she has nothing Greg wants. But Crystal has lots of timber, which Bob, the carpenter, wants. In fact, Bob values Laura's timber more than he values a dental exam. So Bob agrees to give Laura his piece of paper from Greg in exchange for timber.
What just happened here?
Greg got the chair he wanted from Bob, and Bob converted his efforts into an "IOU" from Greg, which he then traded to Laura for timber, who then received a dental exam from Greg. In fact, if Greg's work is highly demanded, than that slip of paper for a Dental exam becomes even more 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 fake those IOU's. Why? Let's say it took Bob 10 hours worth of time/effort/labor to earn 1 IOU from Greg. That IOU now equates in value to 10 hours of Bob's work. So, if Eggert wants a dental exam from Greg, but has nothing to trade for an IOU, the ability to counterfeit an IOU unfairly provides Eggert the opportunity to falsely convert his efforts into something valuable. After all, if no one stops Eggert from doing what he's doing, why would anyone actually earn a real IOU from Greg (which requires actual effort)?
Are you confused yet?
This is the reason why currencies have largely relied on stuff from the real world that is scarce or finite. How so? Take gold, for example. Gold is rare and, at this point of time can only be generated naturally and mined. 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 Greg to give half an exam to one person and half an exam to another.
So, when Bob makes a chair for Greg, Greg 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 the limited natural quantities.
For this reason, Countries began to centralize their gold reserves and, instead, issued 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 among 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.
Fast forwarded a little, 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 generated $20 worth of value with labor, 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.
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 ability to duplicate information has made it difficult making currency, because the second the currency becomes popular, it becomes a target to replicate instead of earning 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 is a limited amount of tokens, they, like gold, provide the important aspect of currency; "How did someone get this?"
In essense, it means that if you own 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 Gregg? If his business closes, 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 because others are willing to trust their value and accept their worth.
The difference between the various Cryptocurrencies comes down to the technology/source/purpose but essentially, they operate to be more attractive, dependable and realiable than others. At the core, they all share the ability to act as legitimate currency.
"Mining" is just like mining for gold. It is the only way to introduce new "stuff" into the system, which has the ability to be a currency.