HBase was inspired by Google's White Paper on Big Table. Big Table's design choice was to provide random access to a high volume of structured and unstructured data.
When you talk about HBase, you should really talk about it as a data store rather than a database. In the traditional sense, a database provides storage capability but also provides certain advanced features such as triggers and secondary indexes. Although HBase has some mechanisms (such as HBase Filters and Co-Processors) to overcome certain limitations of traditional databases, it is still primarily a data store.
HBase is an important component of the Hadoop ecosystem, which leverages the fault-tolerance feature of HDFS.