The basic classification of a form object is a class that contains writable attributes, validations and logic to persist the attributes to ActiveRecord objects. These forms can also include other side-effects like background job triggers, emails, and push-notifications etc. The simplest way to understand the concept is to think of them as a representation of a controller action where all of the business logic that happens in that controller action is abstracted into a form object.
This definition may be a bit too broad. Others (like Reform) define it to have smaller scope — only the part where it persists/validates attributes. The other side effects might be better to put in a different location, like the controller action, or a service/processor object that has a form object.