How can I know if my recipient has opened/read the email that I sent?
By far the easiest and most reliable way to know if your recipient has read a message is to ask them in the message to send you a brief reply.
While Google Apps (for business and government) supports read receipts, Gmail's web interface does not support automatic read receipts or message disposition notifications (MDNs).
But I really want an automatic notification.
Some mail clients allow you to set a special header that requests an automatic receipt notification from the receiving mail client. While you can't do this in Gmail's web interface, you can use a desktop mail client that supports this feature (such as Mozilla Thunderbird or MS Outlook) to send your mail via Gmail's SMTP servers.
There are also variety of web-based services that can be used with any email address to send tracked mails. You should think very carefully before using such services, because they often use shady techniques to track emails. For example, they frequently use techniques otherwise employed only by spammers. Your recipients may consider such techniques to be an attempt to violate their privacy.
What's the problem with read receipts?
There are essentially two problems with read receipts in general:
They are unreliable: For a read receipt to work, the receiving client must support it and be configured to use it. Many clients (including most webmail services) will simply ignore any read receipt request. For this reason, not receiving a read receipt doesn't mean that the mail was not read. In addition, some systems may reply to read receipts without the user having truly opened the message. So receiving a read receipt doesn't necessarily mean the mail was read. In other words, read receipts tell you almost nothing. Even the sneakier techniques used by mail tracking services are far less than perfectly reliable.
They may be considered a violation of privacy: Your recipient may not want you to know exactly when they opened your email. For this reason, most modern email clients won't send a read receipt without the explicit confirmation of the recipient.
I don't care about that. I still think Gmail should have read receipts.
Feel free to use the Suggestion Form to let them know.