If you are seeing @shopify.com appearing in your customer email notifications, or if your notification emails are bouncing back, then you will need to create an SPF record within your domain. Doing so should resolve both of these issues within 24-48 hours of adding the record.
v=spf1 include:shops.shopify.com ~all is the official SPF record that Shopify provides. You can add this to your domain by creating a new "txt" record and inputting the @ symbol under the name column.
If you already have an SPF record, and it is required by one of your other services (such as your email provider), then you'll want to combine the records. For example: v=spf1 include:shops.shopify.com include:otherdomain.com ~all
If you are using Gsuite as your email host and using a TXT record to verify your ownership of the domain, then you would need to create a second TXT record with the value: v=spf1 include:shops.shopify.com include:_spf.google.com ~all.
Our help document here outlines how to go about editing the DNS settings of your Shopify domain. However, if you're using a third-party domain, and are unsure of how to add a "txt" record, then it would be best to reach out to your domain provider. Once the record is added, it will take between 24-48 hours to propagate fully. You can use tools such as MxToolbox to look up the domain name, and check to see if the "txt" record appears.
Keep in mind that you will want to ensure the email you've added to the customers' email field in Settings/General is a valid email with hosting that can send out emails. For example, using @shopname.zendesk.com isn't able to send outgoing email, so if you have that in Settings/General, then it's not going to work - the issue is with the email itself, not Shopify sending emails.
Our support may be limited in what we can help with if you're using a third-party domain; however, if you do run into any issues, or have any questions, we'd welcome you to reach out to us via Facebook & Twitter (@shopifysupport), or through phones or chats.