Sadly there is no standard for the name of such an email - all your suggestions are used. But consider the following:
Verification Email - used when you can still access services, but need to verify your email in the meantime.
Activation Email - used when services (account) are not accessible until email activation takes place. You can argue that there is some more urgency in 'activating' than in 'verifying'.
Confirmation Email - I prefer to think of this as something sent as a confirmation of something (purchase confirmation, password change confirmation). You can say that the email is to confirm the user's email address.
Here are a few examples to get you started (starting with my favourite):
The Noun Project
Please Verify your Noun Project account
Very minimalistic design.
email subject and content terms don't match. Subject says 'Verify' while content says 'Activate'.
No need for the 'Activate Account' heading. It is redundant to the button, which is the primary action point.
Link contrast somewhat low.
Activate your MailChimp account.
Monkey and "Just one more step..." make the system appear more fun and the sentence should motivate people to click on the button.
No need for the line asking people to click on the button. Most people should get it without these instructions (and even if they won't get it - people often try the most probable possible action when inconclusive).
Few will care for the address of the company, or the copyright clause at this point and user eyes will most often skip this. The address of the company does promote trust, but such soft goal is likely to have already been satisfied prior registrations.
No links in case the email was sent by mistake, or if people have problems when clicking the button.
Please confirm your email
Again, no real need for the sentence next to the button.
Two exclamation marks can be considered style that is not ideal. Exclamation marks should be used sparingly for a very specific desired effect.
The other answers deal well with what the email should say, but I wanted to share a nifty feature that is currently available in Gmail (I think you have to apply to use it at the moment).
The problem is that users don't like confirming their email address -- it is just another chore. Gmail has come up with a neat solution so that users do not have to actually open the email. Instead, you just show a 'Confirm' button in the subject line. This is how Mailchimp uses the feature:
You can create one of these buttons by adding some Schema.org markup to your HTML email.
These 'Inbox actions' are still in their infancy, but I like them a lot. They save the user from ever having to view the email to perform obvious and repetitive actions.
Case: User has registered on your website/portal/application and Now before you allow him access, you need him to verify the provided e-mail address.
As per my view I would suggest following:
Subject should say something like "Activate your <Name of your website/application/portal> account"
The Activation link: I as a user don't like the strange and big URL's that some forums publish in the activation e-mails. I would suggest you to add a Button directed to the URL with a text "Activate Here" / "Activate Account".
The content/design of e-mail. I would like to see something like the sample images posted by "Izhaki" in the e-mail.