Straight from the horse's mouth:
Hadoop is a framework for running
applications on large clusters built
of commodity hardware. The Hadoop
framework transparently provides
applications both reliability and data
motion. Hadoop implements a
computational paradigm named
Map/Reduce, where the application is
divided into many small fragments of
work, each of which may be executed or
reexecuted on any node in the cluster.
In addition, it provides a distributed
file system (HDFS) that stores data on
the compute nodes, providing very high
aggregate bandwidth across the
cluster. Both Map/Reduce and the
distributed file system are designed
so that node failures are
automatically handled by the
Map/Reduce is a programming paradigm that was made popular by Google where in a task is divided in to small portions and distributed to a large number of nodes for processing (map), and the results are then summarized in to the final answer (reduce). Google and Yahoo use this for their search engine technology, among other things.
Hadoop is a generic framework for implementing this kind of processing scheme. As for why it kicks ass, mostly because it provides neat features such as fault tolerance and lets you bring together pretty much any kind of hardware to do the processing. It also scales extremely well, provided your problem fits the paradigm.
You can read all about it on the website.
As for some examples, Paul gave a few, but here's a few more you could do that are not so web-centric:
- Rendering a 3D film. The "map" step distributes the geometry for every frame to a different node, the nodes render it, and the rendered frames are recombined in the "reduce" step.
- Computing the energy in a system in a molecular model. Each frame of a system trajectory is distributed to a node in the "map" step. The nodes compute the the energy for each frame,
and then the results are summarized in the "reduce" step.
Essentially the model works very well for a problem that can be broken down in to similar discrete computations that are completely independent, and can be recombined to produce a final result.