A custom field is field specific to your organisation that you may require in addition to the "core" fields.
You can add your own Custom Fields via Organisation Settings > Custom Fields beyond the standard fields TidyHQ provides.
Custom Fields could be a Registration Number, First Aid Certificate Number, School Attended etc.
Each Custom Field has a Title, a Type and an "Allow users to change this field through their profile page" checkbox (that when checked means that the field will be available to use on a public facing Membership Level or Event.
You may also have a variety of options for your custom fields limiting the chance of errors when inputting data. Options are:
Custom Fields can be used for Event Ticket purchases to collect Event or Attendee specific information.
If you want to use Custom Fields to collect information from Contacts, then the "Allow users to change this field through their profile page" must be checked. As you control where the Custom Fields are used, even though this option is selected, it won't appear on all forms etc.
The 'Allow users to change this field through their profile page' option lets you decide if you want the custom field to be displayed on the Contact's Contact Profile. If you choose to display the custom field on the profile, the Contact will have the ability to update it at anytime. This will automatically update the information on your side too.