How do I install and mount an exFAT partition?

You will need the packages exfat-fuse and exfat-utils to allow you to read and write on exFAT drives:

For 14.04 and higher:

sudo apt-get install exfat-fuse exfat-utils

Afterwards you just need to plug in the HDD and you're good to go.

13.10 and later

You will need the exfat-fuse and exfat-utils packages. Install them with:

sudo apt-get install exfat-fuse exfat-utils  

Your exFAT formatted drives should now (auto) mount fine.