Here's how you can explain bitcoin and blockchain to your grandma. Finally :)
Here's a very simplified explanation of how the blockchain (and bitcoin transactions) work - in an analogy.
Blockchain
Imagine that any transaction you make is stored in a public accounting system. Not your name, but an address with the transaction amount that is owned by you or an entity (you pay someone to own it) - therefore fairly anonymous - whilst public. Once the transaction is written in to the system, it can't be changed, because it's copied to tens of thousands of computers around the world. If you want to falsify the transaction on one computer, it will be verified against other computers and by vote of majority, your fake transaction will be deemed invalid and ignored. Very safe system.
Because it doesn't involve any third parties like banks, payment processors (VISA, MasterCard etc), it's very fast - you send the transaction - it gets verified against the system - and arrives to it's destination within minutes or hours (not days like with banks).
This is how blockchain works.
Mining
Imagine that instead of one accountant that writes down the transactions from example above, each participating computer is an accountant. How do we decide which one gets to write the transaction? And how do we incentivize them to do the work?
First - how do we decide which one gets to write the transaction into the accounting system (blockchain)? We let them compete against each other. It's like having many accountants compete to do your accounts at home.
Second - how do we incentivize them to compete? We give the winner - who gets to write the transaction into the blockchain - a reward in the form of a few bitcoins and the small fees that we, ones who transact, pay for the use of the accounting system.
That's basically what's called mining. There's no actual mining involved. It's competition computing - solving a cryptographic puzzle by sheer computing power.
Cryptocurrency (running on blockchain)
A cryptocurrency, like bitcoin runs on this accounting system (blockchain) and is basically a form of sending value from one participant to another. Once a transaction is sent - it is posted to the network and added to a block of transactions. Then, the previously mentioned bunch of accountants (miners) compete to be allowed to write all the transactions in one block into the chain of previous blocks. It's all segregated into blocks, because having it all in one block would be too massive and heavy - and slow. At home or at work you also have different folders, not one massive folder that never ends. That's where the name blockchain comes from.
Bitcoin itself isn't a coin, in the simplest way of saying it - it's just some code. (I might get killed for saying that, but that's what it boils down to. It's actually not that many lines of code. It's a very short software.) Its price goes up and down based on price elasticity - more people want to buy it - price goes up. Demand goes down - price goes down.
Wallet
A wallet is a software that allows you to enter the bitcoin network or the blockchain, send or receive transactions or store received bitcoins. It is a graphical user interface that runs on your PC or mobile phone, or on an USB drive. It is secured usually by a password and a seed - a recovery key in case you lose your password.
Exchanges
They are like stock exchanges, but you just buy coins on them. You send in some fiat money (USD, EUR, GBP, JPY....) and exchange them for cryptocurrencies. Or you sell cryptocurrencies and exchange them back for EUR and then withdraw to your bank account.
You send money in usually via SEPA or WIRE transfer, or credit cards (higher fees). Or you can send in cryptocurrencies (faster option) from your wallet on your PC / mobile / USB to a wallet assigned to your account on chosen exchange.
All together now!
I want to buy a bitcoin and send half a bitcoin to Joe. Here's what I do:
I open an account in an exchange
I send a bunch of my money to the exchange via SEPA from my standard bank account and buy bitcoin for that
I withdraw the bitcoin to my wallet by simply sending it from one address to my wallet address and it arrives in a few minutes
I open my wallet and send half a bitcoin to Joe - he has given me his wallet address (it's just a long unique piece of jibberish, like v4a8sd9tb9f8b4g98bwdsf9gads9fasd687vca)
Joe receives it in an hour (sometimes there's more transactions and the network is a bit slower. But still faster than a bank.
And Joe can either keep the bitcoin or he can reverse the process:
Joe opens an account on an exchange
he sends his bitcoin there
he sells it for USD
he withdraws the USD to his bank account
Of course the exchange to and from USD will not be needed once we can pay online or in the shops using bitcoin too.
Now you (and your grandma) knows, how bitcoin and blockchain works :).