Go the Jobs section on the dashboard view of the group containing the form, this is the last panel on the dashboard view.
Click on the option to Create/edit notifications for this group. You will be taken to the notification pop up.
From here you can click to 'Create notification'. Then fill in the fields required, specifying the task as being 'Document Form Response', select the name of the form and that you want to send an email, and then choose a person you want to receive the notification. If you have no users in this group you can navigate to the root group to choose your user.
Click on Ok, and on Save on the next pop up and the member chosen will then receive an email to notify them when the form is next submitted.
The same process applies if you are setting up a notification to let someone know when there is an application to join the site, but you need to select slightly different options – for 'Group Application' and 'Send an email', and you will only be able to select from users who have permissions in the web office, in order for them to process the application, unlike form responses where you can select to send the notification to anyone who is in your database.