It's a distributed ledger, basically a list of data that everyone has a copy of.
Imagine an island on which 3 people live. Rather than have a bank look after their money, they instead decide to use their own currency and call it islandcoin (IC). The simplest solution would be for one of these people to have the list of every transaction and add to it when someone wants to transfer money from one person to another. So the list would look like:
1st April Alice starts with 10 IC
1st April Bob starts with10 IC
1st April Charlie starts with 10 IC
2nd April Alice transfers 1IC to Bob
3rd April Charlie transfers 2IC to Alice
This is the essence of the blockchain, a list of data.
The problem is however that the 3 people don't trust each other and think that whoever holds the list will just put up fake transactions and steal all their IC.
So instead they decide they will all hold an identical copy of the list and only agree to add a transaction to it if 2 or more out of the 3 agree (50%+ of the participants) that both the transaction is valid. (If it required 100% then essentially one person could veto all transactions, and if it only required one then they would be in the same position as before).
This is what is known as a private blockchain, with a limited number of pre-approved participants.
Crpytocurrencies are public blockchains, meaning anyone can join and run the software. The problem now is that someone could make run the program simultaneously to get over the 50% threshold and steal everyones money. To stop this "mining" is introduced, which essentially means you have to solve a very difficult and pointless problem on your computer, before any data is added to the blockchain. In the case of Bitcoin this has quickly grown to requiring the energy of a medium sized country, to process a small number of transactions per day.
I'm skeptical that Blockchain will be used in more than a few highly niche areas. It's not actually a new idea, it's been around in slightly different formats for decades, yet no one uses it because it's a very inefficient database, hence only usable in quite unique circumstances.