MySQL Logging Options
Well, each log is for a different purpose.


Binary logs - Required for acting as a replication master and useful for auditing or point in time restores from backups
Relay logs - Required for acting as a replication slave. They are only stored until applied to the slave and then removed automatically
General query log - All the queries executed by the server
Slow query log - Queries that are 'slow'. Based on executing time and/or index usage
Innodb logs - These are internal to the innodb engine and not a classic log for you to look at. It stores the data post the current checkpoint and so it can be merged into sequential writes rather then random writes all over the table.
Error logs - Just stderr from the mysqld process


What are you after? Queries run or slow queries or?