How to rsync a directory to a new directory with different name?

If you want to use rsync to recursively make the dest directory an exact copy of the src directory:

rsync -a src/ dest

The rsync man page explains how this works:

A trailing slash on the source [...] avoid[s] creating an additional directory level at the destination. You can think of a trailing / on a source as meaning "copy the contents of this directory" as opposed to "copy the directory by name" [...]