What is Ethereum? What is Ethereum mining?

In the Ethereum universe, there is a single, canonical computer (called the Ethereum Virtual Machine, or EVM) whose state everyone on the Ethereum network agrees on. Everyone who participates in the Ethereum network (every Ethereum node) keeps a copy of the state of this computer. Additionally, any participant can broadcast a request for this computer to perform arbitrary computation. Whenever such a request is broadcast, other participants on the network verify, validate, and carry out ("execute") the computation. This execution causes a state change in the EVM, which is committed and propagated throughout the entire network.

Requests for computation are called transaction requests; the record of all transactions and the EVM's present state gets stored on the blockchain, which in turn is stored and agreed upon by all nodes.

Cryptographic mechanisms ensure that once transactions are verified as valid and added to the blockchain, they can't be tampered with later. 

How does Ethereum mining work?

Ethereum currently uses a proof-of-work consensus mechanism. This means that anyone who wants to add new blocks to the chain must solve a difficult puzzle that requires a lot of computing power. Solving the puzzle "proves" that you have done the "work" by using computational resources. Doing this is known as mining. Mining is typically brute force trial and error, but successfully adding a block is rewarded in ETH.