Technically, Ethereum is a decentralized consensus system where programs with agreement about their state and results of their computation. They are called contracts, but the term doesnt entirely match what they do. The ethers as cryptocurrency are a necessary side effect. Even if it weren't, conversely, distributed consensus plus turing completeness implies cryptocurrencies can be build ontop.
However points of view differ more on what it is to people. It is...
.. a system to make contracts more clear
Unsurprisingly, it is often lawyers with this foremost in their mind. They deal most with regular contracts. From what i have heard (from Casey Kuhlman) is that in contract disputes it is usually about either:
- what the terms mean
- what happened
The code always runs the same, and the consensus mechanism forces agreement on the input of the contract. I.e. about what happened.
.. for DAOs
Decentralized Autonomous Organizations, or Decentralized Organizations, the latter term formed because the autonomy of the contract may be disputable. There is a spectrum between those. When the group behind the organization is tightly nit, and they have a lot of leeway with the contract, it is more like a DO. When it is more loosely knit, or not knit at all, it is a DAO.
I personally like the idea of open DAOs that accept work from anyone, as i espoused in an earlier post.
Both DOs and DAOs may have self-modification mechanisms. (DOUG being one in development) If they do, the mechanisms for disallowing/allowing changes to the DAO have to be effective. For instance a DAO may have stakeholders vote, or to introduce changes on which may be voted. (use of the votes doesnt have to be a simple tally)
Stakeholders could be anything; employees, customers, stock holders, or even other DAOs.(and combinations) People behind these will usually have to thoroughly inspect the change.
The importance of the checks on those modifications are particularly important if the DAO is 'infrastructure like'.
Another point about DAOs/DOs is that they need a business model or social reason for existence. In general contracts need source of income, because running them costs a little ether.
I would be amiss not to mention Vitalics series of DACs, which may have have inspired Ethereum.
.. a tool for derivatives and hedging
People from this point of few seem relatively few in ethereum, they might have been attracted to Mastercoin, CounterParty, colored coins, or Ripple who serve their needs, at least to a large extent.
.. the consensus part of web 3.0
Perhaps the most general one. The idea here is that it will come with a browser, and decentralized collection of files people can add to. These files could be complete webpages, like those representing the above ideas.
One of the founders of Ethereum, Gavin Wood, most strongly comes from this direction. (presentation about it) I think he is the origin of the idea, or at least, that specific clear form. (can't tell for sure)
The "it can run arbitrary programs, it could do anything" comes over as a cop-out, so people dont say that. Instead they say the aspect that is is more in their minds or hearts.
It is a good thing, because only by going down one of the roads can the bandwidth of a conversation reach any detail.
That doesnt mean they dont know about different directions. Some things, like reputation systems, are not in this list because they're largely parts of the above approaches. However, maybe they can be viewed in a way that says something about what it is to people, so the above list may not be exhaustive.