You can find the available integrations under your project's Settings ➔ Integrations page. You need to have at least maintainer permission on the project.
Integrations allow you to integrate GitLab with other applications. They are a bit like plugins in that they allow a lot of freedom in adding functionality to GitLab.
Project webhooks allow you to trigger a URL if for example new code is pushed or a new issue is created. You can configure webhooks to listen for specific events like pushes, issues or merge requests. GitLab will send a POST request with data to the webhook URL.