Trigger-based emails can be generated when a recipient takes an action
such as opening an email, clicking on a link or submitting a form.
Additionally an email can be triggered if a customer or subscriber’s
profile changes or matches a particular demographic. Triggers can be
used for a variety of scenarios such as to automate sales lead
notifications, confirmation emails, send thank you messages for
completing a survey or to send a follow-up offer to a recipient that
clicks on a specific link. (source: EmailLabs)

The Palmer Web Marketing Blog talks about 2 great ways to increase conversions on your website through automated trigger based email messages:

1. Email Customers Who Abandon Their Shopping Carts
An extremely smart tactic to bring back lost sales is to checkup on those people who never complete the checkout process. If you have already captured the customers’ email address, you can easily send them a friendly message informing them that you’re currently trying to improve your checkout process, and you apologize if they found anything confusing. In addition, you might want to give them a coupon code as an additional incentive for bringing them back.

2. Add a "email me when in-stock" feature
This feature is a no-brainer for sites that frequently sell out of their inventory. However, it’s amazing how few websites actually do it. Here’s how it works: Anytime a customer visits a product detail page that displays an "out of stock" notice, give them the option to be emailed if the item becomes available again. This serves two purposes. 

First, it gives you an opportunity to capture their email address (opt in, of course). And second, it gives you the opportunity to save the sale if you re-stock the item. Now wait a minute, you think, most websites don’t allow people to visit pages of items that are out of stock. That’s true, but keep in mind that search engines will keep these pages indexed for some time. In addition, people frequently bookmark pages of products they would like to purchase in the future. This feature is even more useful if you sell a product that comes in multiple sizes or colors. For example, if you sell widgets that are available in both green and blue, and the green widgets are out of stock, your customer’s can be emailed when that color is re-stocked. Just like that, another lost sale is captured!

Source: Palmer Web Marketing Blog