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There are quite a few alternatives and I've actually written a short book on the subject entitled "Online GIS - Meet the Cloud Publication Platforms that Will Revolutionize our Industry" but that's a little outdated now.
Here's an updated summary:
MangoMap: Very easy to use, no coding required. Lots of tools and functionality available to make really polished map applications. Much more competitive pricing than ArcGIS Online organisational accounts.
GISCloud: Online alternative to traditional client/server GIS setup. Many features but hampered by a frustrating user interface.
MapBox: Making maps sexy again. Programmer focused. Great for maps that need to fit a brand and be able to scale for high traffic. Good fit for consumer internet sites.
CartoDB: Attractive UX and scales very well. Also lets you preserve the Google Maps experience for end users. Postgres + postgis database on the cloud with a set of API's on top of it to fetch/save and render data.
Disclosure: Original answer posted by Founder of MangoMaps and includes an edit by the CTO of CartoDB - these two products are described in this answer.