Ethereum is often described as "the world computer".
But what does that mean? Let's start with a computer science–focused description, and then try to decipher that with a more practical analysis of Ethereum’s capabilities and characteristics, while comparing it to Bitcoin and other decentralized information exchange platforms (or "blockchains" for short).
From a computer science perspective, Ethereum is a deterministic but practically unbounded state machine, consisting of a globally accessible singleton state and a virtual machine (EVM) that applies changes to that state.
From a more practical perspective, Ethereum is an open source, globally decentralized computing infrastructure that executes programs called smart contracts. It uses a blockchain to synchronize and store the system’s state changes, along with a cryptocurrency called ether to meter and constrain execution resource costs.

The term "Turing complete" refers to English mathematician Alan Turing, who is considered the father of computer science. In 1936 he created a mathematical model of a computer consisting of a state machine that manipulates symbols by reading and writing them on sequential memory (resembling an infinite-length paper tape). With this construct, Turing went on to provide a mathematical foundation to answer (in the negative) questions about universal computability, meaning whether all problems are solvable. He proved that there are classes of problems that are uncomputable. Specifically, he proved that the halting problem (whether it is possible, given an arbitrary program and its input, to determine whether the program will eventually stop running) is not solvable.
A system is said to be Turing complete if it can be used to simulate any Turing machine. Such a system is called a Universal Turing machine (UTM).
Ethereum's ability to execute a stored program, in a state machine called the Ethereum Virtual Machine, while reading and writing data to memory makes it a Turing-complete system and therefore a UTM. Ethereum can compute any algorithm that can be computed by any Turing machine, given the limitations of finite memory.
Ethereum's groundbreaking innovation is to combine the general-purpose computing architecture of a stored-program computer with a decentralized blockchain, thereby creating a distributed single-state (singleton) world computer. Ethereum programs run "everywhere", yet produce a common state that is secured by the rules of consensus.

Hearing that Ethereum is Turing complete, you might arrive at the conclusion that this is a feature that is somehow lacking in a system that is Turing incomplete. Rather, it is the opposite. Turing completeness is very easy to achieve; in fact, the simplest Turing-complete state machine known has 4 states and uses 6 symbols, with a state definition that is only 22 instructions long. Indeed, sometimes systems are found to be "accidentally Turing complete." A fun reference of such systems can be found at http://bit.ly/2Og1VgX.
However, Turing completeness is very dangerous, particularly in open access systems like public blockchains, because of the halting problem we touched on earlier. For example, modern printers are Turing complete and can be given files to print that send them into a frozen state. The fact that Ethereum is Turing complete means that any program of any complexity can be computed by Ethereum. But that flexibility brings some thorny security and resource management problems. An unresponsive printer can be turned off and turned back on again. That is not possible with a public blockchain.

Turing proved that you cannot predict whether a program will terminate by simulating it on a computer. In simple terms, we cannot predict the path of a program without running it. Turing-complete systems can run in "infinite loops", a term used (in oversimplification) to describe a program that does not terminate. It is trivial to create a program that runs a loop that never ends. But unintended never-ending loops can arise without warning, due to complex interactions between the starting conditions and the code. In Ethereum, this poses a challenge: every participating node (client) must validate every transaction, running any smart contracts it calls. But as Turing proved, Ethereum can’t predict if a smart contract will terminate, or how long it will run, without actually running it (possibly running forever). Whether by accident or on purpose, a smart contract can be created such that it runs forever when a node attempts to validate it. This is effectively a DoS attack. And of course, between a program that takes a millisecond to validate and one that runs forever are an infinite range of nasty, resource-hogging, memory-bloating, CPU-overheating programs that simply waste resources. In a world computer, a program that abuses resources gets to abuse the world’s resources. How does Ethereum constrain the resources used by a smart contract if it cannot predict resource use in advance?
To answer this challenge, Ethereum introduces a metering mechanism called gas. As the EVM executes a smart contract, it carefully accounts for every instruction (computation, data access, etc.). Each instruction has a predetermined cost in units of gas. When a transaction triggers the execution of a smart contract, it must include an amount of gas that sets the upper limit of what can be consumed running the smart contract. The EVM will terminate execution if the amount of gas consumed by computation exceeds the gas available in the transaction. Gas is the mechanism Ethereum uses to allow Turing-complete computation while limiting the resources that any program can consume.
The next question is, 'how does one get gas to pay for computation on the Ethereum world computer?' You won't find gas on any exchanges. It can only be purchased as part of a transaction, and can only be bought with ether. Ether needs to be sent along with a transaction and it needs to be explicitly earmarked for the purchase of gas, along with an acceptable gas price. Just like at the pump, the price of gas is not fixed. Gas is purchased for the transaction, the computation is executed, and any unused gas is refunded back to the sender of the transaction.

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What is Ethereum? - Mastering Ethereum
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What is Ethereum?
Ethereum and Turing Completeness
Turing Completeness as a "Feature"
Implications of Turing Completeness