Wait a few minutes/hours and your domain should be working (it can take up to 48 hours).
Please also note that if you have forwarding setup or have perviously setup your domain to point to another host this must be undone / turned off in order to work with Salon.io.
On your site settings page on Salon.io enter your domain as well as a title you want to appear at the top of the browser. You can also add a google tracking ID or some meta tags.
Double click on the @ record or click the pencil and edit the address to read 126.96.36.199 then click save zone file.
Find the '@' record. If it doesn't exist create it and if any others exist remove them. It should look just like this page.
Open the zone file editor by clicking the launch button.
Find the domain you want to link to salon and click the launch button that corresponds to it on the right.
Login to your go daddy account then go to my account in the top left.
GoDaddy custom domain setup guide
The following guide specifically deals with the registrar www.godaddy.com however it is likely that the process is similar on other registrar's sites.