HDFS is a distributed filesystem; One can do most regular FS operations on it such as listing files in a directory, writing a regular file, reading a part of the file, etc. Its not simply "a collection of structured or unstructured data" anymore than your EXT4 or NTFS filesystems are.
HBase is a in-memory Key-Value store which may persist to HDFS (it isn't a hard-requirement, you can run HBase on any distributed-filesystem). For any read key request asked of HBase, it will first check its runtime memory caches to see if it has a value cached, and otherwise visit its stored files on HDFS to seek and read out the specific value. There are various configurations in HBase offered to control the way the cache is utilised, but HBase's speed comes from a combination of caching and indexed persistence (faster, seek-ed file reads).
HBase's file-based persistence on HDFS does the key indexing automatically when it writes, so there is no manual indexing need by its users. These files are regular HDFS files, but specialised in format for HBase's usage, known as HFiles.
These articles are slightly dated, but are still very reflective of the architecture HBase uses: http://blog.cloudera.com/blog/2012/06/hbase-write-path/ and http://blog.cloudera.com/blog/2012/06/hbase-io-hfile-input-output/, and should help if you want to dig deeper.
