Ethereum, a virtual currency, enables transactions that rival Bitcoins
nytimes.com
nytimes.com
In addition, it runs your service robustly, and preserves the entire history of the computation forever. The downside is that these transparency and robustness features make the service a lot more expensive than ordinary cloud computing, as well as more difficult to program on.
It doesn't really have much to do with Bitcoin or currency (except that (a) like Bitcoin, Ethereum uses blockchain technology to achieve these things; but that's an implementation detail (b) a Bitcoin clone is one of the many things you could build as an Ethereum application (c) for technical reasons you buy Ethereum server time via purchasing server-time-credits called "Ether").
As a project, Ethereum is very technically ambitious (they are not just working on a cloud computing platform, but also trying to advance blockchain technology in general, eg developing a new proof-of-stake algorithm, and eg developing a way to not have to replicate computations on every node on the network). They are trying to do a lot of new things at once, which is risky (but awesome).