Web development has always been about distributed computing, since clients have been on different machines to the servers they talk to, web pages can pull in resources from many servers to build a page's content, and servers may talk to other machines to achieve their goals. CDNs make this more obvious than before, but really they're just an evolution, an introduction of a virtualization/indirection layer between what you ask for and the hardware used to provide it.
Clouds are about taking the concepts of virtualization and applying them to remote hosting, both of low-level OSes and higher-level software platforms. The really interesting thing about them is that this enables different business models on the part of customers (and with different risks too, but that's mostly not related to the fact that it's distributed computing but rather that it is not wholly under your control in your own jurisdiction).
I've found that the most effective use of distributed computing is when you think in terms of connecting together distinct services, each of which with different capabilities (which might be for technical reasons, or might not; sometimes, it's for business or legal reasons that things have to be divided up) and where each of those services may be provided by many components in multiple locations. There are, and continue to remain, issues with balancing the need for performance (which is a force that brings components together) and the need for robustness (which tends to lead to distribution and replication) within the overall context of the general capabilities map.
My goodness! That paragraph sounds like terrible piffle! What I'm trying to say is that it's all trade-offs, and you should be prepared for not getting it right first time.
(Hadoop is a mechanism for doing a distributed file store, and for efficiently applying certain classes of operation – those that fit well with MapReduce or other similar scatter-gather algorithms – across that whole dataset. If that shoe fits, use it. But it doesn't solve all problems, and thank goodness for that! Things that can do everything tend to look very much like things that can't actually do anything at all, and usefulness and comprehensibility come in the restrictions.)