The first generation of "internet hard drive" companies (xdrive, idrive, etc) were based on the old LAN model of a file share that happens to live on the Internet. Accessing every file over the internet was slow, creating a ton of friction.
Dropbox provided true synchronization technology, so performance was fast and you could continue to work on files offline.
Opinions vary, but I think one of the key ingredients in their success was the "it just works" factor.
Dropbox took a seemingly simple issue that people had struggled with forever (easy sharing of files over a network) and made it a transparent process.
I think it also helped that the service didn't also co-market itself as a shady way to share copyright material (eg: the peer-to-peer networks of yore that claimed they were for easy sharing, but just happened to be filled with movies, porn, and mp3's).
Dropbox is an excellent example of finding a common pain point for consumers and solving that one specific problem in a highly efficient and effective manner.
Drew Houston actually has presentation on this called Startup Lessons Learned. It will run you through most of what you want to know.
When I got hooked on Dropbox there weren't really that many alternatives to it. There were a few that offered a backup + some sort of syncing interface on top of Amazon S3 and there was Microsoft's Live Mesh which was in beta and didn't look very appealing at the time.
Here is why I personally think DB was successful.
Dropbox is a very simple tool and simplicity is very attractive when it comes to a complex problem like folder syncing. Dropbox is also totally painless. It works on Windows, Linux and Mac. Read about some more reasons here http://www.itworld.com/software/235425/more-reasons-why-dropbox-has-won-so-many-geeky-hearts