Yes, you can buy stock in 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 advantage of Ethereum over Bitcoin, is that it can support many different types of decentralized applications.
The most important thing to know before buying stock in Ethereum, is that it is still in development and the price is highly volatile. So buying stock in Ethereum is only for investors who are comfortable with risk and are willing to hold for the long term.