How do I know whether an email was accepted by the recipient's mail server?

You usually can’t know for sure from the moment you send it. What you can know depends on what level of “accepted” you mean:

1) Accepted by the receiving server

If you got a successful SMTP response during delivery, such as:

• 250 OK
• 250 2.0.0 queued as ...
• 250 message accepted

then the recipient’s mail server accepted the message for delivery.

2) Delivered to the user’s inbox

That’s different. A server may accept the email and then later:

• put it in spam/junk
• quarantine it
• silently drop it
• bounce it later if delivery fails downstream

So SMTP acceptance does not mean the recipient read it or even saw it.

3) What you can rely on

• Immediate SMTP success from the recipient server = accepted by that server.
• Bounce message / DSN = not accepted or later rejected.
• No bounce = not proof of delivery, just that no failure was reported yet.

4) For mail you send via your own server

If you have access to mail logs, look for:

• the recipient server’s SMTP reply code
• whether your server marked the message as accepted/queued/sent
• any later bounce or deferred status

5) Best practical indicators

• Positive delivery confirmation: server replied 250
• Failure confirmation: bounce/NDR from the receiving domain or your mail system
• Read confirmation: only if the recipient client supports/read-receipt and the user allows it, which is unreliable

If you want, I can also show you how to interpret SMTP reply codes or how to check this in Exchange/Postfix/Gmail logs.