(While the question is broad, I hope that an equally broad answer will spark a useful discussion.)
Please see the accepted answer for What is Hadoop?What is Hadoop? at SO.
Google searches are a question that Hadoop can answer. Think about the characteristics of a large search engine:
- Large amounts of data
- Distributed data
- Extreme parallelism
Scalability was mentioned in the comments: With Hadoop, it is not hard to throw additional (commodity) servers into the mix.
On to your question. If your project has a lot of SQL and von Neumann bottlenecks, then Hadoop makes little sense. If, however, your data is "Big Data," is less structured, and may be parallelized, then Hadoop will make more sense.