What does "Drop Connections" and "Update Statistics" in the "Detach Database" dialog do?

Drop Connections will force the database into Single User mode before detaching the database, thereby forcing all users to disconnect. If you don't check-mark that option, and users are connected to the database, the database will not be detached.

Update Statistics will refresh any out of date statistics before detaching the database. If you leave this option un-checked, statistics objects will remain as they were at the time you detached the database.

Microsoft has an excellent article with more details on MSDN.

Supplementing to the current answer...

Detach database will leave you without any backup. If for some reason, you cannot attach database, you will end up with a detached copy of mdf and ldf.

A best practice should be to do a backup and restore of the database. This way you end up with at least a copy of the database.

Read up : http://blogs.sqlsentry.com/aaronbertrand/bad-habits-file-backups/