Ethereum is the network, also known as the blockchain. Ether (ETH) is the fuel for that network. When you send tokens, interact with a contract, send ETH, or do anything else on the blockchain, you must pay for that computation. That payment is calculated in Gas and gas is paid in ETH.
You are paying for the computation, regardless of whether your transaction succeeds or fails. Even if it fails, the miners must validate and execute your transaction (compute) and therefore you must pay for that computation just like you would pay for a successful transaction.
You can see your TX fee (gas limit * gas price) in ETH & USD when you search for your transaction on etherscan.io. This is not a TX fee that MyEtherWallet, or any other service provider, receives. This fee is paid to miners for mining transactions, putting them into blocks, and securing the blockchain.
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