Amazon SES wants to make sure that you're able to most successful in sending your messages, and provides you feedback when emails you send are bounced or are invalid. It does this by supplying a "Rejected" status for your Campaign Members, which is useful to know, but not the easiest to debug the reason why it was set as "Rejected".
Here's a few troubleshooting tips when a Campaign Member has a "Rejected" status.
1) Confirm that the sender's email address is a verified sender in Amazon SES.
2) Make sure the sender's email is entered accurately. Note that it is case sensitive, and must appear exactly correct.
3) If you're still in the testing mode with Amazon SES (not in production mode), confirm that the address you're sending to is verified in Amazon SES.
4) Confirm that your email has content in both the HTML Body and Text Body versions.
5) Confirm that your chosen Email Template includes the {!Campaign.EmailBodyHTML} merge tag in it.
6) If you're using the Nonprofit Success Pack (NPSP) from Salesforce.org, ensure that the Lead or Contact you've targeted for your message has actually populated the Lead.Email or Contact.Email field. Often times in testing, these fields may be left black due to short-cutting the NPSP processes you could be using during testing, and thereby sending the email to a blank address.
7) Are you able to login to your Amazon Web Services account? If not, your Amazon Web Services account may be blocked, stopping Soapbox Mailer from being able to communicate with Amazon. If you're unable to login to your Amazon Web Services account, it would be good to make sure your account is in good billing standing with Amazon, and if so, contact Amazon Web Services.
8) Have you reached your 24-hour email sending quota in Amazon SES? You can find this by logging into your Amazon SES account directly to see both your quota as well as how many emails you've sent in the past 24 hours. If your limit has been reached, all emails you try to send until that limit is decreased will result in a "communications failure" between Salesforce and Amazon.
If you're still have troubles, open a support ticket and we'll get things squared away!