Hadoop is targeted for developing programs to process large data sets. It's useful whenever you have a lot of data to process or analyze. The first Hadoop application for many web companies is to analyze log data. For example, you can look at log data to see how many unique viewers you have and where do they tend to come from. Another popular usage is to analyze user data to understand them better.
One of the original inspiration for Hadoop/MapReduce is crawling and indexing for search engines. In fact, Doug Cutting, the inventor of Lucene, is also the inventor of Hadoop.