How to I check the delivery status of my client communications?

For an individual client, you can check this in their Email Log (Contacts > Contact Name > Email Log) by looking at the "Status" column. You will see whether the communication was delivered, open, if links were clicked, or if the message bounced/dropped and was not delivered.

For group emails and newsletters, you can look at the Reports menu > Email/Newsletter section > Email & Newsletters Stats report, and for autoresponders in the Autoresponder Stats of the same reports section.