I don't get what is that? It's id which auto increase number?
An index in a database is an aid to help searching for information within specific columns of a database table. But while an index helps searching, it also impacts data being inserted, updated and/or deleted.
It's id which auto increase number?
No, that's not an index - it's a sequence value, used in MySQL so that when you add records to a table, the column value is always unique. Because the value is unique, searching by it will always return the same, single record.
And index is a structure that the database server manages which helps it sort and search a table faster. There are a ton of related concepts you'll want to read up on (Binary Trees, Linear searching, lots more) to get up to speed, but start with http://en.wikipedia.org/wiki/Index_(database) for an overview.
Indexes are basically ways for a DB to speed up a search for data. When you have tons of records it really help to be able to look up certain data from a query faster. You just have to be aware of how to use them and when they don't work like you might think!
Auto incrementing numbers are a feature you can use to have MySQL generate unique IDs for a column.