Back in the Pardot days, we wanted a way to accept data from a third-party form or application directly into the app, so we came up with the form handler. Basically, it’s the receiving end of a form submission without the front end form fields to go with it. Over time, this practice of sending and receiving data via a specific URLs became popular and is widely know as a webhook.
Wikipedia has a more technical definition:
Webhooks are “user-defined HTTP callbacks”. They are usually triggered by some event, such as pushing code to a repository or a comment being posted to a blog. When that event occurs, the source site makes an HTTP request to the URI configured for the webhook. Users can configure them to cause events on one site to invoke behaviour on another.
Now, webhooks make it easy to share data and trigger features across thousands of SaaS apps, and they don’t have to have formal integrations. The more apps that implement webhooks — both to send and receive data — the more the incumbent apps that aren’t SaaS 2.0 lose their ecosystem integration moat.
Look for webhooks to continue to grow in importance over time.
What else? What are some more thoughts on the power of webhooks?
David Cummings on Startups 
My experience has been that it’s hard to justify business value of building webhooks into a web platform. Much of the value that comes from webhooks is from the ease of integration that future customers/partners will have, which is very hard to quantify on a spreadsheet. In my current role, we allow http polling to our platform which is significantly harder to maintain on the side of the integration implementer than webhook based integrations would be. Since a viable workaround exists, webhooks themselves never get prioritized (squeaky tire gets the oil).
Another important thing to consider with webhooks is that they are a contract that your company is making. Your company will need to maintain its webhooks in their current form forever or risk upsetting customers by breaking their custom integrations. As your platform evolves, it can make it very hard to maintain a legacy webhook, so weight that careful before building them (or any public api)