A cryptocurrency such as Bitcoin, Ethereum, Litecoin or Monero is a decentralized digital currency that allows you to use blockchain technology to buy goods and services. They come in many different names, but essentially they will use an online ledger to store and secures all transactions with strong cryptography. It can be thought of like checkbook that is distributed across many computers around the world. Each new transaction is recorded in 'blocks' that are linked together with previous transactions on a 'chain' and logged within the global ledger.
To buy cryptocurrencies, you will need to setup a 'wallet' for the type of coin you want to use. The wallet is an online app that can hold your currency. Generally, you will create an account on an exchange, and then you can transfer real money to buy cryptocurrencies. Coinbase is a cryptocurrency trading exchange where you can create a wallet to buy and sell many types of cryptocurrencies.
With the blockchain, everyone who uses the cryptocurrency is provided a copy of the global ledgar to ensure a unified transaction record. Software will log each new transaction as it happens, and every copy of the blockchain is updated simultaneously with the new information, keeping all records identical and accurate across all wallets and users. Many wallets will have to 'sync' transactions to ensure you have all missing transactions.
Some companies are issuing their own currencies, often called tokens, which can be traded specifically for goods or service that the specific company provides. Think of them as you would arcade tokens or casino chips. You will need to exchange a real currency for a cryptocurrency coin on a specific network to access the good or service.