This was a great video intro to the blockchain.
In short, blockchain is a immutable linked list of hashes. Which can be used to solve the double spending problem in cryptocurrencies using proof of work and a persistent chain of records (blocks mined).
PoW: The more hashing power the more trust worthy.
PoS: The more coins the more trust worthy.
PBFT: Hand selected trusted nodes.
Randomized asynchronous algorithm: You trust nobody! Every single node is constantly re-evaluated. (Skycoin)
Horizon State - Digital voting.
Blockstack - Decentralized apps on Blockstack let you own your data so you can maintain your privacy, security and freedom.
Cardano - Cryptographic currency implementing Ouroboros PoS protocol.
Iden3 - Decentralized identity management solution.