Yes, you can buy a piece of Ethereum. Ethereum is a decentralized platform that runs smart contracts: applications that run exactly as programmed without any possibility of fraud or third party interference.
Ethereum is not just a platform but also a programming language (Turing complete) running on a blockchain that helps developers to build and publish distributed applications. The code of these applications is running on the Ethereum network and therefore has to be bought in order to be used.
If you want to buy a piece of Ethereum, the easiest way is to use an online exchange that accepts fiat currency (like USD or EUR) in exchange for Ethereum. Once you have bought your ETH, you can store it in a wallet that supports the ERC20 standard, such as MyEtherWallet or MetaMask.
Ethereum is often described as a digital currency or cryptocurrency, but it is more accurately described as a decentralized platform that runs smart contracts. These contracts are executed by the Ethereum Virtual Machine (EVM), which runs on the Ethereum network.
The EVM executes code exactly as programmed, without any possibility of fraud or third party interference.