I recommend you have a listen to the TAC Security Podcast #32. It had an excellent talk all about ASA syslogs and how to use them. You can find it (and all the other episodes) here.
Syslogs are much more than just connection and access-list records, although they can include those too. You basically set severity levels you are interested in and send the logs to a log server where they are archived and searchable (and perhaps setup to take certain action such as alert an admin if something goes wrong).
Many people set their logging level too high and then proceed to not use the syslog since a firewall with logging level 6 will create a message for every single tcp session establishment and teardown. That can often mean hundreds of thousand or even millions of message per day.
If you want an overview of the logging levels and how to setup logging in general, refer to this section of the configuration guide.
