You can use cp with the -r (copy recursively) flag and specify the new directory name:
cp -r /path/to/directory /path/to/location/new-name
In the above command replace the given paths with the real ones.
For example, to copy stuff from my home directory to an existing directory named backup and name the new directory stuff-backup (if this directory already exists, note that stuff will be copied into it, not overwrite it), you run:
cp -r ~/stuff ~/backup/stuff-backup
~ is a shortcut for your home directory /home/$USER or /home/zanna in my case. You can omit it if the current working directory is your home directory - you can see this from your prompt
zanna@monster:~$
^--the ~ here is where I am - home!
You can add the -v (verbose) flag to make cp report each copy being performed:
$ cp -vr stuff backup/stuff-backup
'stuff/thing1' -> 'backup/stuff-backup/thing1'
'stuff/thing2' -> 'backup/stuff-backup/thing2
...