I realize this is a very old question, but I thought I would offer an analogy to help illustrate the fine answers above.
If you walk into a public library, you will find that the books are all arranged in a particular order (most likely the Dewey Decimal System, or DDS). This corresponds to the "clustered index" of the books. If the DDS# for the book you want was 005.7565 F736s, you would start by locating the row of bookshelves that is labeled 001-099 or something like that. (This endcap sign at the end of the stack corresponds to an "intermediate node" in the index.) Eventually you would drill down to the specific shelf labelled 005.7450 - 005.7600, then you would scan until you found the book with the specified DDS#, and at that point you have found your book.
But if you didn't come into the library with the DDS# of your book memorized, then you would need a second index to assist you. In the olden days you would find at the front of the library a wonderful bureau of drawers known as the "Card Catalog". In it were thousands of 3x5 cards -- one for each book, sorted in alphabetical order (by title, perhaps). This corresponds to the "non-clustered index". These card catalogs were organized in a hierarchical structure, so that each drawer would be labeled with the range of cards it contained (Ka - Kl, for example; i.e., the "intermediate node"). Once again, you would drill in until you found your book, but in this case, once you have found it (i.e, the "leaf node"), you don't have the book itself, but just a card with an index number (the DDS#) with which you could find the actual book in the clustered index.
Of course, nothing would stop the librarian from photocopying all the cards and sorting them in a different order in a separate card catalog. (Typically there were at least two such catalogs: one sorted by author name, and one by title.) In principle, you could have as many of these "non-clustered" indexes as you want.