Ethereum is a decentralized platform that runs smart contracts. That will run applications exactly as programmed without any possibility of downtime, censorship, fraud or third party interference.
These apps run on a custom built blockchain. It´s an enormously powerful shared global infrastructure. The blockchain can move value around and represent the ownership of property. This enables developers to create markets, store registries of debts or promises, move funds in accordance with instructions. The instructions can be given long in the past (like a will or a futures contract) and many other things that have not been invented yet. All this can happen without a middle man or counterparty risk.
The Ethereum Wallet is a gateway to decentralized applications on the Ethereum blockchain. It allows you to hold and secure ether and other crypto-assets built on Ethereum, as well as write, deploy and use smart contracts.
The currency unit of the Ethereum platform is called Ether. It is traded on cryptocurrency exchanges like any other cryptocurrency. It´s used inside Ethereum platform to pay for computational services on the network.