Indexes is a method which database systems use to quickly find data. The real world analogy are indexes in books. If an author/publisher does a good job at indexing their book, it becomes pretty easy for the reader to directly go to the page they want to read simply by looking at the index. Same goes for a database. If an index is created on a field, the database pre-sorts the data. When a request is made on the data, the database uses the index to identify which location the data is stored in on the hard disk, and directly goes there. If there are no indexes, the database needs to look at every record in order to find out if it meets the criteria(s) of your query.
A simple way to look at indexes is by thinking of a deck of cards. A database which is not indexed is like a deck a cards which have been shuffled. If you want to find the king of spades, you need to look at every card one by one to find it. You might be lucky and it can be the first one, or you might be unlucky and it can be the last one.
A database which is indexed, has all the cards in the deck ordered from ace to king and each suite is set aside in its own pile. Looking for the king of spades is much simpler now because you simply need to look at the bottom of the pile of cards which contains the spades.
I hope this helps. Be warned though that although indexes are necessary in a relational database system, they can counter productive if you write too many of them. There's a ton of great articles on the web that you can read up on indexes. I'd suggest doing some reading before you dive into them.
