If you're technically minded, the Bitcoin whitepaper is a good starting point. Then you can read about how subsequent cryptocurrencies improve, or attempt to improve, on the original design. If that's too much, I'll give a quick overview.
You know BitTorrent, right? On BitTorrent, a bunch of people all have the same file and agree on its contents. When someone new comes along, they can download parts of the file from anyone/everyone and know it's trustworthy, because all the group can prove they have parts of a file with the same contents. And if anyone tries to lie about a part of the file, they can't, because the contents of the file are proven via cryptographic hash.
Bitcoin is a similar idea, except instead of a file, it's a transaction ledger. Each new transaction is "confirmed" by hashing it with the previous transaction in a way that's easy to verify but time-consuming to compute in the first place ("Miners" are the people that do the time-consuming computing). Then the next transaction is confirmed on top of that one, and so on. This chain of transactions is the "blockchain". "Block" comes from that in the real protocol, unconfirmed transactions are grouped into a big list, or "block", and hashed with the previous block, instead of one at a time.
The benefit of this is that you rapidly reach a trusted consensus on transactions across the globe that can't be faked or changed, without needing a trusted intermediary like Paypal or Visa, similar to BitTorrent not needing a trusted server to download a file. This gives you a mathematically verifiable "coin" that can't be counterfeited, relies on no central authority, and is guaranteed to behave a certain way.
: Basically, treat some data as a very long number and enter it into an equation that produces a fixed-size result that is unlikely to occur for anything besides that exact number, and can't be "reversed" to see what the possible inputs are. Thus, you can verify the data that was entered into the equation. For example: take this post and assign each letter a number from 1 to 26. Then add up all the numbers. Then take the first letter of each sentence and do the same thing. Then append the two numbers. That sequence of numbers is unlikely to occur for any other post, and you can't take the numbers and figure out the contents of the post.