What is Ethereum?

What is Ether?
Ether is a necessary element -- a fuel -- for operating the distributed application platform Ethereum. It is a form of payment made by the clients of the platform to the machines executing the requested operations. To put it another way, ether is the incentive ensuring that developers write quality applications (wasteful code costs more), and that the network remains healthy (people are compensated for their contributed resources).

How are ethers created?
The total supply of ether and its rate of issuance was decided by the donations gathered on the 2014 presale. The results were roughly:

60 million ether created to contributors of the presale
12 Million (20% of the above) were created to the development fund, most of it going to early contributors and developers and the remaining to the Ethereum Foundation
5 ethers are created every block (roughly 15-17 seconds) to the miner of the block
2-3 ethers are sometimes sent to another miner if they were also able to find a solution but his block wasn't included (called uncle/aunt reward)
Is the ether supply infinite?
No. According to the terms agreed by all parties on the 2014 presale, issuance of ether is capped at 18 million ether per year (this number equals 25% of the initial supply). This means that while the absolute issuance is fixed, the relative inflation is decreased every year. In theory if this issuance was kept indefinitely then at some point the rate of new tokens created every year would reach the average amount lost yearly (by misuse, accidental key lost, death of holders etc) and there would reach an equilibrium.

But the rate is not expected to be kept: sometime in 2017 Ethereum will be switched from Proof of Work to a new consensus algorithm under development, called Casper that is expected to be more efficient and require less mining subsidy. The exact method of issuance and which function it will serve is an area of active research, but what can be guaranteed now is that (1) the current maximum is considered a ceiling and the new issuance under casper will not exceed it (and is expected to be much less) and (2) whatever method is ultimately picked to issue, it will be a decentralized smart contract that will not give preferential treatment to any particular group of people and whose purpose is to benefit the overall health and security of the network.

Who needs ether?
Developers who intend to build apps that will use the ethereum blockchain. Users who want to access and interact with smart contracts on the ethereum blockchain.