What is double opt-in or verified opt-in?
The subscribers interest is confirmed second time by clicking on a link that he actually wants to receive email from you. In other words the subscribers interest is verified twice.
Is it annoying for the subscriber?
It is not a problem for the subscriber to confirm his interest if he *is* really interested in your list. It will become your problem if you don't set your mailing list software in this double opt-in mode.
Why? See the example below ...
PersonA entered an email address of PersonB into your subscription form. You didn't set your autoresponder to process double opt-ins. You are sending emails to this email address to PersonB who doesn't know anything about you. He thinks you are spamming. He might blacklist your mail server ip or report to antispam authorities about your spamming.
You don't know you are spamming ... but because of the missed double opt-in step, you will be in trouble.
How to enable double opt-in or verified opt-in?
Whether it is a third party autoresponder service or a autoresponder script installed on your site, you should be able to see an option like this in the admin panel. You need to select that option for the double opt-in to be kicked when someone subscribed.

What are the advantages of verified opt-in?
* Tight targeted list: Subscribers are more interested in your emails as they went an extra step of confirming to get your emails.
* You are out of trouble: While they are confirming, your software gets their ip, time of subscription (your autoresponder should be able to record these) and the web site from which they are subscribed. So you have the proof that they actually confirmed their subscription.
* Invalid email addresses: Subscriber has to enter the actual email that he/she is using to get the confirmation link, so there are more chances that the email address will be used by him in the future.
Any drawbacks if you use double opt-in?
Internet marketers are little scared that double opt-in affects the development of their list. Because subscribers might forget to confirm their subscription.
But think about it ... 100 subscribers who are confirmed telling you that they are interested in your emails are way better than 1000 subscribers who entered your list with single opt-in. So it is always adviced that an email marketer should use double opt-in to develop their lists.